1. Introduction

Let $\Omega \subset {\mathbb{R}^3}$ denote an open and bounded domain whose smooth boundary $\Gamma$ has the outward unit normal n; let $T>0$ be a given final time, and set $Q:=\Omega\times (0, T)$ and $\Sigma:=\Gamma\times (0, T)$. We study in this paper distributed optimal control problems of the following form:

(CP) Minimize the cost functional

$J(u, \rho, \mu)=\frac{{{\beta _1}}}{2}\int_Q {} |\rho-{\rho _Q}{|^2}dxdt + \frac{{{\beta _2}}}{2}\int_Q {} |\mu-{\mu _Q}{|^2}dx{\mkern 1mu} dt + \frac{{{\beta _3}}}{2}\int_Q {} |u{|^2}dx{\mkern 1mu} dt$

(1.1)

subject to the state system

$(1 + 2{\mkern 1mu} g(\rho)){\mkern 1mu} {\partial _t}\mu + \mu {\mkern 1mu} g'(\rho){\mkern 1mu} {\partial _t}\rho-\Delta \mu=u{\mkern 1mu} \quad a.{\mkern 1mu} e.in{\mkern 1mu} Q,$

(1.2)

${\partial _t}\rho {\mkern 1mu} + {\mkern 1mu} B[\rho]{\mkern 1mu} + {\mkern 1mu} F'(\rho)=\mu {\mkern 1mu} g'(\rho){\mkern 1mu} \quad a.{\mkern 1mu} e.in{\mkern 1mu} Q,$

(1.3)

${\partial _{\text{n}}}\mu=0{\mkern 1mu} \quad a.{\mkern 1mu} e.on\Sigma,$

(1.4)

$\rho (\cdot, 0)={\rho _0}{\mkern 1mu}, \quad \mu (\cdot, 0)={\mu _0}, {\mkern 1mu} \quad a.{\mkern 1mu} e.in{\mkern 1mu} \Omega,$

(1.5)

and to the control constraints

$u \in {U_{{\text{ad}}}}:=\{ u \in {H^1}(0, T;{L^2}(\Omega)){\mkern 1mu} :{\mkern 1mu} \; 0 \leq u \leq {u_{{\text{max}}}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} a.{\mkern 1mu} e.in{\mkern 1mu} Q{\mkern 1mu} {\text{ and }}{\mkern 1mu} {\left\| u \right\|_{{H^1}(0, T;{L^2}(\Omega))}}{\mkern 1mu} \leq {\mkern 1mu} R\}.$

(1.6)

Here, \gianni{$\beta_1, \beta_2, \beta_3\geq0$ and $R>0$ are given constants, with $\beta_1 + \beta_2 + \beta_3 >0$, } and the threshold function ${u_{{\text{max}}}} \in {L^\infty }(Q)$ is nonnegative}. Moreover, $\rho_Q, \mu_Q\in L^2(Q)$ represent prescribed target functions of the tracking-type functional $J$. Although more general cost functionals could be admitted for large parts of the subsequent analysis, we restrict ourselves to the above situation for the sake of a simpler exposition.

The state system (1.2)-(1.5) constitutes a nonlocal version of a phase field model of Cahn-Hilliard type describing phase segregation of two species (atoms and vacancies, say) on a lattice, which was recently studied in ^[18]. In the (simpler) original local model, which was introduced in ^[25], the nonlocal term B[$\rho$] is replaced by the diffusive term-Δ$\rho$. On the other hand, it is important from the point of view of applications to have a nonlocal operator (see, e.g., the Introduction of ^[18]). Indeed, terms of double integral type are more natural in the free energy, whereas squared gradients can be seen as limiting situations of nonlocal contributions.

The local model has been the subject of intensive research in the past years; in this connection, we refer the reader to ^[4-7,9-12]. In particular, in ^[8] the analogue of the control problem (CP) for the local case was investigated for the special situation g($\rho$)=$\rho$, for which the optimal boundary control problems was studied in ^[14].

The state variables of the model are the order parameter $\rho$ and the chemical potential $\mu$. While $\rho$ can be interpreted as a volumetric density, the chemical potential $\mu$ plays as the coldness in the entropy imbalance (see [25, formulas (2) and (24)]). Hence, we must have 0≤$\rho$≤1 and $\mu$ > 0 almost everywhere in Q. The control function u on the right-hand side of (1.2) has the meaning of a microenergy source. We remark at this place that the requirement encoded in the definition of U_ad, namely that u be nonnegative, is indispensable for the analysis of the forthcoming sections. Indeed, it is needed to guarantee the nonnegativity of the chemical potential $\mu$.

The nonlinearity F is a double-well potential defined in the interval (0, 1), whose derivative F′ is singular at the endpoints $\rho$=0 and $\rho$=1: e. g., F=F₁ + F₂, where F₂ is smooth and

${F_1}(\rho)=\hat c{\mkern 1mu} (\rho {\mkern 1mu} \log (\rho) + (1-\rho){\mkern 1mu} \log (1-\rho)), {\mkern 1mu} {\text{with}} {\text{a}} {\text{constant}} {\mkern 1mu} \hat c > 0.$

(1.7)

The presence of the nonlocal term B[$\rho$] in (1.3) constitutes the main difference to the local model and introduces some difficulties in the mathematical analysis due to the lack of compacness and less regularity for the solution. Simple examples are given by integral operators of the form

$B[\rho](x, t)=\int_{\Omega \times (0, t)} k (t, s, x, y){\mkern 1mu} \rho (y, s){\mkern 1mu} dy{\mkern 1mu} ds$

(1.8)

and purely spatial convolutions like

$B[\rho](x, t)=\int_Q {} k(|y-x|){\mkern 1mu} \rho (y, t){\mkern 1mu} dy,$

(1.9)

with sufficiently regular kernels.

Optimal control problems of the above type often occur in industrial production processes. For instance, consider a metallic workpiece consisting of two different component materials that tend to separate. Then a typical goal would be to monitor the production process in such a way that a desired distribution of the two materials (represented by the function $\rho _Q$) is realized during the time evolution in order to guarantee a wanted behavior of the workpiece; the deviation from the desired phase distribution is measured by the first summand in the cost J. The third summand of J represents the costs due to the control action u; the size of the factors $\beta_i\ge 0$ then reflects the relative importance that the two conflicting interests“realize the desired phase distribution as closely as possible”and“minimize the cost of the control action”have for the manufacturer.

The state system (1.2)-(1.5) is singular, with highly nonlinear and nonstandard coupling. In particular, unpleasant nonlinear terms involving time derivatives occur in (1.2), and the expression F′($\rho$) in (1.3) may become singular. Moreover, the nonlocal term B[$\rho$] is a source for possible analytical difficulties, and the absence of the Laplacian in (1.3) may cause a low regularity of the order parameter $\rho$. We remark that the state system (1.2)-(1.5) was recently analyzed in ^[18] for the case u=0 (no control); results concerning well-posedness and regularity were established.

The mathematical literature on control problems for phase field systems involving equations of viscous or nonviscous Cahn-Hilliard type is still scarce and quite recent. We refer in this connection to the works ^{[2,3,16,17,21,28]}. Control problems for convective Cahn-Hilliard systems were studied in ^[29,30], and a few analytical contributions were made to the coupled Cahn-Hilliard/Navier-Stokes system (cf. ^{[19,20,22,23]}). The very recent contribution ^[13] deals with the optimal control of a Cahn-Hilliard type system arising in the modeling of solid tumor growth.

The paper is organized as follows: in Section 2, we state the general assumptions and derive new regularity and stability results for the state system. In Section 3, we establish the directional differentiability of the control-to-state operator, and the final Section 4 brings the main results of this paper, namely, the derivation of the first-order necessary conditions of optimality.

Throughout this paper, we will use the following notation: we denote for a (real) Banach space X by ${\left\| \cdot \right\|_X}$ its norm and the norm of X×X×X, by X′ its dual space, and by ${\left\langle { \cdot, \cdot } \right\rangle _X}$ the dual pairing between X′ and X. If X is an inner product space, then the inner product is denoted by ${\left({ \cdot, \cdot } \right)_X}$. The only exception from this convention is given by the L^p spaces, 1≤p≤∞, for which we use the abbreviating notation ${\left\| \cdot \right\|_p}$ for the norms. Furthermore, we put

$H:={L^2}\left(\Omega \right), \quad V:={H^1}(\Omega), \quad W:=\{ w \in {H^2}(\Omega):{\mkern 1mu} {\partial _{\text{n}}}w=0{\mkern 1mu} {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}} {\text{on}} \Gamma \}.$

We have the dense and continuous embeddings $W\subset V\subset H\cong H'\subset V'\subset W'$, where $\langle u, v\rangle_V=(u, v)_H$ and $\langle u, w\rangle_W=(u, w)_H$ for all $u\in H$, $v\in V$, and $w\in W$.

In the following, we will make repeated use of Young’s inequality

$a{\mkern 1mu} b \leq \delta {\mkern 1mu} {a^2} + \frac{1}{{4\delta }}{b^2}\quad {\text{for all }}{\mkern 1mu} a, b \in \mathbb{R}{\mkern 1mu} \quad {\text{and }}{\mkern 1mu} \delta > 0,$

(1.10)

as well as of the fact that for three dimensions of space and smooth domains the embeddings $\, V\subset L^p(\Omega)$, $1\le p\le 6$, and $\, H^2(\Omega)\subset C^0(\overline{\Omega})$ are continuous and (in the first case only for $1\le p < 6$) compact. In particular, there are positive constants $\widetilde K_i$, $i=1, 2, 3$, which depend only on the domain $\Omega$, such that

${\left\| v \right\|_6} \leq {\widetilde K_1}{\mkern 1mu} {\left\| v \right\|_V}\quad \forall {\mkern 1mu} v \in V,$

(1.11)

${\left\| {v{\mkern 1mu} w} \right\|_H} \leq {\left\| v \right\|_6}{\mkern 1mu} {\left\| w \right\|_3} \leq {\widetilde K_2}{\left\| v \right\|_V}{\left\| w \right\|_V}\quad \forall {\mkern 1mu} v, w \in V,$

(1.12)

${\left\| v \right\|_\infty } \leq {\widetilde K_3}{\left\| v \right\|_{{H^2}(\Omega)}}\quad \forall {\mkern 1mu} v \in {H^2}(\Omega).$

(1.13)

We also set for convenience

${Q_t}:=\Omega \times (0, t){\text{ and }}{Q^t}:=\Omega \times (t, T), \quad {\text{for }}{\mkern 1mu} t \in (0, T).$

(1.14)

Please note the difference between the subscript and the superscript in the above notation.

About time derivatives of a time-dependent function $v$, we point out that we will use both the notations ${\partial _t}v, \, \partial _t^2 v$ and the shorter ones $v_t, \, v_{tt}$.

---

2. Problem statement and results for the state system

Consider the optimal control problem (1.1)-(1.6). We make the following assumptions on the data:

(A1) F=F₁ + F₂, where $F_1\in C^3(0, 1)$ is convex, $F_2\in C^3[0,1]$, and

$\lim_{r\searrow 0} F_1'(r)=-\infty, \quad \lim_{r\nearrow 1} F_1'(r)=+\infty.$

(2.1)

(A2) $\rho_0\in V$, ${F'} (\rho_0) \in H$, $\mu_0\in W$, where $\, \mu_0\ge 0$\, a.\, e. in $\Omega$,

$\inf\, \{\rho_0(x): \, x\in\Omega\}>0, \quad \sup\, \{\rho_0(x):\, x\in\Omega\}<1\, .$

(2.2)

(A3) $g\in C^3[0,1]$ satisfies $g(\rho)\geq 0$ and $g''(\rho)\le 0$ for all $\rho\in [0,1].

(A4) The nonlocal operator $B\colon L^1(Q)\to L^1(Q)$ satisfies the following conditions:

(i) For every $t\in (0, T]$, we have

$B[v]|_{Q_t}=B[w]|_{Q_t} \, \, \mbox{ whenever }\, v|_{Q_t}=w|_{Q_t}.$

(2.3)

(ii) For all $p\in [2, +\infty]$, we have $B(L^p(Q_t))\subset L^p(Q_t)$ and

${\left\| {B[v]} \right\|_{{L^p}({Q_t})}}{\mkern 1mu} \leq {C_{B, p}}\left({1 + {{\left\| v \right\|}_{{L^p}({Q_t})}}} \right)$

(2.4)

for every $v\in L^p(Q)$ and $t\in(0, T]$.

(iii) For every $v, w\in L^1(0, T;H)$ and $t\in (0, T]$, it holds that

$\int_0^t {} {\left\| {B[v](s)-B[w](s)} \right\|_6}d s \leq {\mkern 1mu} {C_B}\int_0^t {} {\left\| {v(s)-w(s)} \right\|_H}d s.$

(2.5)

(iv) It holds, for every $v\in L^2(0, T;V)$ and $t\in (0, T]$, that

${\left\| {\nabla B[v]} \right\|_{{L^2}\left({0, t;H} \right)}} \leq {C_B}(1 + {\mkern 1mu} {\left\| v \right\|_{{L^2}\left({0, t;V} \right)}}).$

(2.6)

(v) For every $v\in H^1(0, T; H)$, we have ${\partial _t}B[v] \in {L^2}(Q)$ and

${\left\| {{\partial _t}B[v]} \right\|_{{L^2}(Q)}} \leq {C_B}(1 + {\left\| {{\partial _t}v} \right\|_{{L^2}(Q)}}).$

(2.7)

(vi) B is continuously Fréchet differentiable as a mapping from L²(Q) into L²(Q), and the Fréchet derivative $DB[\overline{v}]\in {\cal L}(L^2(Q), L^2(Q))$ of $B$ at $\overline{v}$ has for every $\overline{v}\in L^2(Q)$ and $t\in(0, T]$ the following properties:

${\left\| {DB[\bar v](w)} \right\|_{{L^p}({Q_t})}} \leq {C_B}{\left\| w \right\|_{{L^p}({Q_t})}}\quad \forall {\mkern 1mu} w \in {L^p}(Q), \quad \forall {\mkern 1mu} p \in [2,6],$

(2.8)

${\left\| {\nabla (DB[\bar v](w))} \right\|_{{L^2}({Q_t})}} \leq {C_B}{\mkern 1mu} {\left\| w \right\|_{{L^2}(0, t;V)}}\quad \forall {\mkern 1mu} w \in {L^2}(0, T;V).$

(2.9)

In the above formulas, C_{B, p} and C_B denote given positive structural constants. We also notice that (2.8) implicitely requires that $DB[\overline v](w)|_{Q_t}$ depends only on $w|_{Q_t}$. However, this is a consequence of (2.3).

The statements related to the control problem (CP) depend on the assumptions made in the Introduction. We recall them here.

(A5) J and U_ad are defined by (1.1) and (1.6), respectively, where

${\beta _1}{\mkern 1mu}, {\mkern 1mu} {\beta _2}{\mkern 1mu}, {\mkern 1mu} {\beta _3} \geq 0, \quad {\text{ }}{\beta _1} + {\beta _2} + {\beta _3} > 0, {\text{ and }}R > 0.$

(2.10)

${\rho _Q}, {\mu _Q} \in {L^2}\left(Q \right), \quad {u_{{\text{max}}}} \in {L^\infty }\left(Q \right){\text{ }} {\text{and}} {\text{ }}{u_{{\text{max}}}} \geq 0\quad {\text{ a}}{\text{.e}}{\text{.}}\; {\text{in}}Q.$

(2.11)

Remark 1: In view of (2.8), for every $t\in [0, T]$ it holds that

${\left\| {B[v]-B[w]} \right\|_{{L^2}({Q_t})}} \leq {C_B}{\left\| {v-w} \right\|_{{L^2}({Q_t})}}\quad \forall {\mkern 1mu} v, w \in {L^2}\left(Q \right){\mkern 1mu},$

(2.12)

that is, the condition (2.9) in ^[18] is fulfilled. Moreover, (2.4) and (2.6) imply that B maps L²(0, T; V) into itself and that, for all $t\in (0, T]$\, and \, $v\in L^2(0, T;V)$,

$\left| {\int_{{Q_t}} {} \nabla B[v] \cdot \nabla vdxds} \right| \leq {C_B}\left({1{\mkern 1mu} + {\mkern 1mu} \left\| v \right\|_{{L^2}\left({0, t;V} \right)}^2} \right),$

which means that also the condition (2.10) in ^[18] is satisfied. Moreover, thanks to (2.8) and (2.9), there is some constant ${\widetilde C_B} > 0$ such that

${\left\| {DB[\bar v](w)} \right\|_{{L^2}(0, T;V)}}{\mkern 1mu} \leq {\widetilde C_B}{\mkern 1mu} {\left\| w \right\|_{{L^2}(0, T;V)}}\quad \forall {\mkern 1mu} \bar v \in {L^2}\left(Q \right), {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall {\mkern 1mu} w \in {L^2}(0, T;V){\mkern 1mu}.$

(2.13)

R_EMARK 2: We recall (cf. ^[18]) that the integral operator (1.9) satisfies the conditions (2.3) and (2.4), provided that the integral kernel k belongs to C¹(0, +∞) and fulfills, with suitable constants C₁ > 0, C₂ > 0, $0 < \alpha < \frac 32$, $0 < \beta < \frac 52$, the growth conditions

$|k(r)|\le C_1\, r^{-\alpha}, \quad |k'(r)|\le C_2\, r^{-\beta}, \quad\forall\, r>0\, .$

In this case, we have $2\alpha < 3$ and thus, for all $v, w\in L^1(0, T;H)$\, and $t\in (0, T]$,

$\begin{gathered} \int_0^t {} {\left\| {B[v](s)-B[w](s)} \right\|_6}ds \hfill \\ \leq {\mkern 1mu} {C_1}\int_0^t {} {(\int_\Omega {} |\int_\Omega {} |y-x{|^{-\alpha }}|v(y, s)-w(y, s)|dy{|^6}dx)^{1/6}}ds \hfill \\ \leq {\mkern 1mu} {C_3}\int_0^t {} {(\int_\Omega {} |{(\int_\Omega {} |y-x{|^{-2\alpha }}dy)^{1/2}}{\left\| {v(s)-w(s)} \right\|_H}{|^6}dx)^{1/6}}ds \hfill \\ \leq {\mkern 1mu} {C_4}\int_0^t {} {\left\| {v(s)-w(s)} \right\|_H}ds, \hfill \\ \end{gathered}$

with global constants C_i, 3≤i≤4; the condition (2.5) is thus satisfied. Also condition (2.6) holds true in this case: indeed, for every $t\in (0, T]$ and $v\in L^2(0, T;V)$, we find, since $6\beta /5 < 3$, that

$\begin{gathered} \left\| {\nabla B[v]} \right\|_{{L^2}\left({0, t;H} \right)}^2 \leq {C_2}\int_{{Q_t}} {} |\int_\Omega {} |y-x{|^{-\beta }}|v(y, s)|dy{|^2}dxds \hfill \\ \leq {\mkern 1mu} {C_5}\int_{{Q_t}} {} {(\int_\Omega {} |y-x{|^{-6\beta /5}}dy)^{5/3}}\left\| {v(s)} \right\|_6^2dxds \hfill \\ \leq {\mkern 1mu} {C_6}\int_0^t {} \left\| {v(s)} \right\|_V^2ds. \hfill \\ \end{gathered}$

Finally, since the operator B is linear in this case, we have $DB[\overline{v}]=B$ for every $\bar v \in {L^2}\left(Q \right)$, and thus also (A4)(v) and (2.8)-(2.13) are fulfilled. Notice that the above growth conditions are met by, e. g., the three-dimensional Newtonian potential, where k(r)=c/r with some c≠0.

We also note that (A2) implies $\mu_0\in C(\overline\Omega)$, and (A1) and (2.2) ensure that both $F(\rho_0)$ and $F'(\rho_0)$ are in L^∞(Ω), whence in H. Moreover, the logarithmic potential (1.7) obviously fulfills the condition (2.1) in (A1).

We have the following existence and regularity result for the state system.

T_HEOREM 2.1: Suppose that (A1)-(A5) are satisfied. Then the state system (1.2)-(1.5) has for every u∈U_ad a unique solution $(\rho, \mu)$ such that

$\rho \in {H^2}(0, T;H) \cap {W^{1, \infty }}(0, T;{L^\infty }\left(\Omega \right)) \cap {H^1}(0, T;V),$

(2.14)

$\mu \in {W^{1, \infty }}(0, T;H) \cap {H^1}(0, T;V) \cap {L^\infty }(0, T;W) \subset {L^\infty }(Q).$

(2.15)

Moreover, there are constants $0 < \rho_* < \rho^* < 1$, $\mu^*>0$, and $K_1^*>0$, which depend only on the given data, such that for every u∈U_ad the corresponding solution $(\rho, \mu)$ satisfies

$0 < {\rho _*} \leq \rho \leq {\rho ^*} < 1, {\mkern 1mu} \quad 0 \leq \mu \leq {\mu ^*}, {\mkern 1mu} \quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} Q,$

(2.16)

$\begin{gathered} {\left\| \mu \right\|_{{W^{1, \infty }}(0, T;H) \cap {H^1}(0, T;V) \cap {L^\infty }(0, T;W) \cap {L^\infty }(Q)}} \hfill \\ + {\mkern 1mu} {\left\| \rho \right\|_{{H^2}(0, T;H) \cap {W^{1, \infty }}(0, T;{L^\infty }\left(\Omega \right)) \cap {H^1}(0, T;V)}}{\mkern 1mu} \leq {\mkern 1mu} K_1^*. \hfill \\ \end{gathered}$

(2.17)

P_ROOF: In the following, we denote by ${C_i} > 0, i \in \mathbb{N}$, constants which may depend on the data of the control problem (CP) but not on the special choice of u∈U_ad. First, we note that in [18, Thms. 2.1, 2.2] it has been shown that under the given assumptions there exists for $u\equiv 0$ a unique solution $(\rho, \mu)$ with the properties

$0 < \rho < 1, \quad \mu \geq 0, \quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} Q,$

(2.18)

$\rho \in {L^\infty }(0, T;V), \quad {\text{ }}{\partial _t}\rho \in {L^6}(Q),$

(2.19)

$\mu \in {H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^\infty }(Q) \cap {L^2}(0, T;{W^{2, 3/2}}(\Omega)).$

(2.20)

A closer inspection of the proofs in ^[18] reveals that the line of argumentation used there (in particular, the proof that $\mu$ is nonnegative) carries over with only minor modifications to general right-hand sides u∈U_ad. We thus infer that (1.2)-(1.5) enjoys for every u∈U_ad a unique solution satisfying (2.18)-(2.20); more precisely, there is some C₁ > 0 such that

${\left\| \mu \right\|_{{H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^\infty }(Q) \cap {L^2}(0, T;{W^{2, 3/2}}(\Omega))}}{\mkern 1mu} + {\mkern 1mu} {\left\| \rho \right\|_{{L^\infty }(0, T;V)}}{\mkern 1mu} + {\mkern 1mu} {\left\| {{\partial _t}\rho } \right\|_{{L^6}(Q)}}{\mkern 1mu} \leq {\mkern 1mu} {C_1}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}.$

(2.21)

Moreover, invoking (2.18), and (2.4) for p=+∞, we find that

${\left\| {B[\rho]} \right\|_{{L^\infty }(Q)}} \leq {C_2}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}},$

and it follows from (2.21) and the general assumptions on $\rho_0$, $g$, and F, that there are constants $\rho_*, \rho^*$ such that, for every u∈U_ad,

$\begin{gathered} 0 < {\rho _*} \leq \inf {\mkern 1mu} \{ {\rho _0}(x):x \in \Omega \} {\mkern 1mu} \leq {\mkern 1mu} \sup {\mkern 1mu} \{ {\rho _0}(x):x \in \Omega \} {\mkern 1mu} \leq {\mkern 1mu} {\rho ^*} < 1, \hfill \\ F’(\rho) + B[\rho]-\mu {\mkern 1mu} g’(\rho){\mkern 1mu} \leq {\mkern 1mu} 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{if}} 0 < \rho \leq {\rho _*}, \hfill \\ F’(\rho) + B[\rho]-\mu {\mkern 1mu} g’(\rho) \geq 0{\mkern 1mu} {\mkern 1mu} {\text{if}} {\mkern 1mu} {\rho ^*} \leq \rho < 1. \hfill \\ \end{gathered}$

Therefore, multiplying (1.3) by the positive part $(\rho-\rho^*)^+$ of $\rho-\rho^*$, and integrating over Q, we find that

$\begin{gathered} 0=\int_Q {} {\partial _t}\rho {\mkern 1mu} {(\rho-{\rho ^*})^ + }dxdt + \int_Q {} (F'(\rho) + B[\rho]-\mu {\mkern 1mu} g'(\rho)){(\rho-{\rho ^*})^ + }dxdt \\ \geq \frac{1}{2}\int_\Omega {} {\left| {{{(\rho (t)-{\rho ^*})}^ + }} \right|^2}dx, \\ \end{gathered}$

whence we conclude that $(\rho-\rho^*)^+=0$, and thus $\rho\le\rho^*$, almost everywhere in Q. The other bound for $\rho$ in (2.16) is proved similarly.

It remains to show the missing bounds in (2.17) (which then also imply the missing regularity claimed in (2.14)-(2.15)). To this end, we employ a bootstrapping argument.

First, notice that (A3) and the already proved bounds (2.21) and (2.16) imply that the expressions $\mu {\mkern 1mu} g'(\rho){\mkern 1mu} {\partial _t}\rho$ and $$(1 + 2g(\rho)){\mkern 1mu} {\partial _t}\mu {\mkern 1mu}$$ are bounded in L²(Q). Hence, by comparison in (1.2), the same holds true for $\Delta\mu\,$, and thus standard elliptic estimates yield that

${\left\| \mu \right\|_{{L^2}(0, T;W)}}{\mkern 1mu} \leq {\mkern 1mu} {C_3}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}.$

(2.22)

Next, observe that (A1) and (2.16) imply that ${\left\| {F'(\rho)} \right\|_{{L^\infty }(Q)}}{\mkern 1mu} \leq {\mkern 1mu} {C_4}$, and comparison in (1.3), using (A3), yields that

${\left\| {{\partial _t}\rho } \right\|_{{L^\infty }(Q)}}{\mkern 1mu} \leq {\mkern 1mu} {C_5}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}.$

(2.23)

In addition, we infer from the estimates shown above, and using (2.6), that the right-hand side of the identity

$\nabla\rho_t=-F''(\rho)\, \nabla\rho-\nabla B[\rho]+g'(\rho)\, \nabla\mu +\mu\, g''(\rho)\, \nabla\rho$

(2.24)

is bounded in L²(Q), so that

${\left\| {{\partial _t}\rho } \right\|_{{L^2}(0, T;V)}}{\mkern 1mu} \leq {\mkern 1mu} {C_6}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}.$

(2.25)

We also note that the time derivative ${\mkern 1mu} {\partial _t}(-F'(\rho)-B[\rho] + \mu g'(\rho)){\mkern 1mu}$ exists and is bounded in L²(Q) (cf. (2.7)). We thus have

${\left\| {{\rho _{tt}}} \right\|_{{L^2}\left(Q \right)}}{\mkern 1mu} \leq {\mkern 1mu} {C_7}\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}.$

(2.26)

At this point, we observe that Eq. (1.2) can be written in the form

$a{\partial _t}\mu + \mu {\partial _t}a-\Delta \mu=b, \quad {\text{with }}{\mkern 1mu} a:=1 + 2g(\rho), \quad b:=u + \mu {\mkern 1mu} g'(\rho){\mkern 1mu} {\partial _t}\rho,$

where, thanks to the above estimates, we have, for every u∈U_ad,

${\left\| a \right\|_{{L^\infty }(Q)}} + {\left\| {{\partial _t}a} \right\|_{{L^\infty }(Q)}} + {\left\| b \right\|_{{L^\infty }(Q)}} \leq {C_8},$

(2.27)

${\left\| {{\partial _t}^2a} \right\|_{{L^2}(Q)}}{\mkern 1mu}={\mkern 1mu} 2{\mkern 1mu} {\left\| {g''(\rho)\rho _t^2 + g'(\rho){\rho _{tt}}} \right\|_{{L^2}(Q)}} \leq {C_9},$

(2.28)

${\left\| {{\partial _t}b} \right\|_{{L^2}(Q)}}={\left\| {{u_t} + {\mu _t}g'(\rho){\rho _t} + \mu g''(\rho)\rho _t^2 + \mu g'(\rho){\rho _{tt}}} \right\|_{{L^2}(Q)}} \leq {C_{10}}.$

(2.29)

Since also $\mu_0\in W$, we are thus in the situation of [15, Thm. 3.4], whence we obtain that ${\partial _t}\mu \in {L^\infty }(0, T;H) \cap {L^2}(0, T;V)$ and $\mu \in {L^\infty }(0, T;W)$. Moreover, a closer look at the proof of [15, Thm. 3.4] reveals that we also have the estimates

${\left\| {{\partial _t}\mu } \right\|_{{L^\infty }(0, T;H) \cap {L^2}(0, T;V)}} + {\left\| \mu \right\|_{{L^\infty }(0, T;W)}} \leq {C_{11}}.$

(2.30)

This concludes the proof of the assertion.

R_EMARK 3: From the estimates (2.16) and (2.17), and using the continuity of the embedding $V\subset L^6(\Omega)$, we can without loss of generality (by possibly choosing a larger $K_1^*$) assume that also

$\begin{gathered} {\max _{0 \leq i \leq 3}}{\left\| {{F^{(i)}}(\rho)} \right\|_{{L^\infty }(Q)}}{\mkern 1mu} + {\mkern 1mu} {\max _{0 \leq i \leq 3}}{\left\| {{g^{(i)}}(\rho)} \right\|_{{L^\infty }(Q)}} + {\left\| {\nabla \mu } \right\|_{{L^\infty }(0, T;{L^6}{{(\Omega)}^3})}} \hfill \\ + {\left\| {{\partial _t}\mu } \right\|_{{L^2}(0, T;V)}}{\mkern 1mu} + {\left\| {B[\rho]} \right\|_{{H^1}(0, T;{L^2}(\Omega)) \cap {L^\infty }(Q) \cap {L^2}(0, T;V)}} \leq {\mkern 1mu} K_1^*\quad \forall {\mkern 1mu} u \in {U_{{\text{ad}}}}{\mkern 1mu}. \hfill \\ \end{gathered}$

(2.31)

According to Theorem 2.1, the control-to-state mapping $S:{U_{{\text{ad}}}} \ni u \mapsto (\rho, \mu)$ is well defined. We now study its stability properties. We have the following result.

T_HEOREM 2.2: Suppose that (A1)-(A5) are fulfilled, and let u_i∈U_ad, i=1, 2, be given and $(\rho_i, \mu_i)={\cal S}(u_i)$, i=1, 2, be the associated solutions to the state system (1.2)-(1.5). Then there exists a contant $K_2^*>0$, which depends only on the data of the problem, such that, for every $t\in (0, T]$,

$\begin{gathered} {\left\| {{\rho _1}-{\rho _2}} \right\|_{{H^1}(0, t;H) \cap {L^\infty }(0, t;{L^6}(\Omega))}}{\mkern 1mu} + {\left\| {{\mu _1}-{\mu _2}} \right\|_{{H^1}(0, t;H) \cap {L^\infty }(0, t;V) \cap {L^2}(0, t;W)}} \hfill \\ \leq {\mkern 1mu} K_2^*{\mkern 1mu} {\left\| {{u_1}-{u_2}} \right\|_{{L^2}(0, t;H)}}{\mkern 1mu}. \hfill \\ \end{gathered}$

(2.32)

P_ROOF: Taking the difference of the equations satisfied by $(\rho_i, \mu_i)$, i=1, 2, and setting $u:=u_1-u_2$, $\rho:=\rho_1-\rho_2$, $\mu:=\mu_1-\mu_2$, we first observe that we have almost everywhere in Q the identities

$\begin{gathered} (1 + 2g({\rho _1})){\partial _t}\mu {\mkern 1mu} + {\mkern 1mu} g'({\rho _1}){\partial _t}{\rho _1}{\mkern 1mu} \mu-\Delta \mu {\mkern 1mu} + {\mkern 1mu} 2(g({\rho _1})-g({\rho _2})){\partial _t}{\mu _2} \hfill \\=u{\mkern 1mu}-{\mkern 1mu} (g'({\rho _1})-g'({\rho _2})){\partial _t}{\rho _1}{\mkern 1mu} {\mu _2}{\mkern 1mu}-{\mkern 1mu} g'({\rho _2}){\mkern 1mu} {\mu _2}{\partial _t}\rho {\mkern 1mu}, \hfill \\ \end{gathered}$

(2.33)

$\begin{gathered} {\partial _t}\rho {\mkern 1mu} + {\mkern 1mu} F'({\rho _1}){\mkern 1mu}-F'({\rho _2}){\mkern 1mu} + {\mkern 1mu} B[{\rho _1}]-B[{\rho _2}] \hfill \\=g'({\rho _1}){\mkern 1mu} \mu {\mkern 1mu} + {\mkern 1mu} (g'({\rho _1})-g'({\rho _2})){\mkern 1mu} {\mu _2}{\mkern 1mu}, \hfill \\ \end{gathered}$

(2.34)

as well as

${\partial _{\text{n}}}\mu=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}} {\text{on}} {\mkern 1mu} \Sigma, \quad \mu (\cdot, 0)=\rho (\cdot, 0)=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}} {\text{in}}{\mkern 1mu} \Omega.$

(2.35)

Let $t\in (0, T]$ be arbitrary. In the following, we repeatedly use the global estimates (2.16), (2.17), and (2.31), without further reference. Moreover, we denote by c > 0 constants that may depend on the given data of the state system, but not on the choice of u₁, u₂∈U_ad; the meaning of c may change between and even within lines. We establish the validity of (2.32) in a series of steps.

S_TEP 1: To begin with, we first observe that

$(1 + 2g({\rho _1}))\mu {\partial _t}\mu + g'({\rho _1}){\partial _t}{\rho _1}{\mu ^2}={\partial _t}((\frac{1}{2} + g({\rho _1})){\mu ^2}).$

Hence, multiplying (2.33) by $\mu$ and integrating over Q_t and by parts, we obtain that

$\int_\Omega {} (\frac{1}{2} + g({\rho _1}(t))){\mu ^2}(t)dx + \int_{{Q_t}} {} |\nabla \mu {|^2}dx ds \leq \sum\limits_{j=1}^3 | {I_j}|{\mkern 1mu},$

(2.36)

where the expressions I_j, j=1, 2, 3, defined below, are estimated as follows: first, we apply (A3), the mean value theorem, and Hälder’s and Young’s inequalities, to find, for every $\gamma>0$ (to be chosen later), that

$\begin{gathered} {I_1}:=-2\int_{{Q_t}} {} (g({\rho _1})-g({\rho _2})){\partial _t}{\mu _2}{\mkern 1mu} \mu dx ds \leq c\int_0^t {} {\left\| {{\partial _t}{\mu _2}(s)} \right\|_6}{\mkern 1mu} {\left\| {\mu (s)} \right\|_3}{\mkern 1mu} {\left\| {\rho (s)} \right\|_2}ds \hfill \\ \leq {\mkern 1mu} \gamma \int_0^t {} \left\| {\mu (s)} \right\|_V^2ds{\mkern 1mu} + {\mkern 1mu} \frac{c}{\gamma }\int_0^t {} \left\| {{\partial _t}{\mu _2}(s)} \right\|_V^2{\mkern 1mu} \left\| {\rho (s)} \right\|_H^2ds, \hfill \\ \end{gathered}$

(2.37)

where it follows from (2.31) that the mapping $s \mapsto \left\| {{\partial _t}{\mu _2}(s)} \right\|_V^2$ belongs to L¹(0, T). Next, we see that

$\begin{gathered} {I_2}:=\int_{{Q_t}} {} (u-(g'({\rho _1})-g'({\rho _2})){\partial _t}{\rho _1}{\mkern 1mu} {\mu _2}){\mkern 1mu} \mu dx ds \hfill \\ \leq c\int_{{Q_t}} {} (|u| + |\rho |)|\mu |dx ds \leq c\int_{{Q_t}} {} ({u^2} + {\rho ^2} + {\mu ^2})dx ds. \hfill \\ \end{gathered}$

(2.38)

Finally, Young’s inequality yields that

${I_3}:=-\int_{{Q_t}} {} g'({\rho _2}){\mkern 1mu} {\mu _2}{\mkern 1mu} {\rho _t}{\mkern 1mu} \mu dx ds \leq \gamma \int_{{Q_t}} {} \rho _t^2dx ds + {\mkern 1mu} \frac{c}{\gamma }\int_{{Q_t}} {} {\mu ^2}dx ds{\mkern 1mu}.$

(2.39)

Combining (2.36)-(2.39), and recalling that $g(\rho_1)$ is nonnegative, we have thus shown the estimate

$\begin{gathered} \frac{1}{2}{\mkern 1mu} \left\| {\mu (t)} \right\|_H^2{\mkern 1mu} + {\mkern 1mu} (1-\gamma)\int_0^t {} \left\| {\mu (s)} \right\|_V^2ds \leq \gamma \int_{{Q_t}} {} \rho _t^2dx ds + {\mkern 1mu} c\int_{{Q_t}} {} {u^2}dx ds \hfill \\ + {\mkern 1mu} c\left({1 + {\gamma ^{-1}}} \right)\int_0^t {} (\left\| {\mu (s)} \right\|_H^2{\mkern 1mu} + {\mkern 1mu} (1 + \left\| {{\partial _t}{\mu _2}(s)} \right\|_V^2){\mkern 1mu} \left\| {\rho (s)} \right\|_H^2)ds. \hfill \\ \end{gathered}$

(2.40)

Next, we add $\rho$ on both sides of (2.34) and multiply the result by $\rho_t$. Integrating over Qt, using the Lipschitz continuity of F′ (when restricted to $[\rho_*, \rho^*]$), (2.12) and Young’s inequality, we easily find the estimate

$(1-\gamma)\int_{{Q_t}} {} \rho _t^2dx ds + \frac{1}{2}{\mkern 1mu} \left\| {\rho (t)} \right\|_H^2 \leq \frac{c}{\gamma }\int_{{Q_t}} {} ({\rho ^2} + {\mu ^2})dx ds.$

(2.41)

Therefore, combining (2.40) with (2.41), choosing $\gamma>0$ small enough, and invoking Gronwall’s lemma, we have shown that

${\left\| \mu \right\|_{{L^\infty }(0, t;H) \cap {L^2}(0, t;V)}} + {\left\| \rho \right\|_{{H^1}(0, t;H)}} \leq c{\left\| u \right\|_{{L^2}(0, t;H)}}\quad \forall {\mkern 1mu} t \in (0, T].$

(2.42)

S_TEP 2: Next, we multiply (2.34) by $\, \rho|\rho|\,$ and integrate over Q_t. We obtain

$\frac{1}{3}{\mkern 1mu} \left\| {\rho (t)} \right\|_3^3{\mkern 1mu} \leq {\mkern 1mu} \sum\limits_{j=1}^3 | {J_j}|,$

(2.43)

where the expressions J_j, 1≤j≤3, are estimated as follows: at first, we simply have

$\begin{gathered} {J_1}{\text{: = }}\int_{{Q_t}} {} ( - F'({\rho _1}) + F'({\rho _2}) + {\mu _2}(g'({\rho _1}) - g'({\rho _2})))\rho |\rho |dxds \\ ≤ c\int_0^t {} \left\| {\rho (s)} \right\|_3^3ds. \\ \end{gathered}$

(2.44)

Moreover, invoking (2.42), Hälder’s inequality, as well as the global bounds once more,

$\begin{gathered} {J_2}:=\int_{{Q_t}} {} \mu {\mkern 1mu} g'({\rho _1}){\mkern 1mu} \rho {\mkern 1mu} |\rho |dx ds \leq c\int_0^t {} {\left\| {\mu (s)} \right\|_6}{\mkern 1mu} {\left\| {\rho (s)} \right\|_2}{\mkern 1mu} {\left\| {\rho (s)} \right\|_3}ds \\ \leq \int_0^t {} \left\| {\rho (s)} \right\|_3^3ds + {\mkern 1mu} c\int_0^t {} \left\| {\mu (s)} \right\|_V^{3/2}\left\| {\rho (s)} \right\|_H^{3/2}ds \\ \leq \int_0^t {} \left\| {\rho (s)} \right\|_3^3ds + c{\mkern 1mu} \left\| \rho \right\|_{{L^\infty }(0, t;H)}^{3/2}\left\| \mu \right\|_{{L^{3/2}}(0, t;V)}^{3/2} \\ \leq \int_0^t {} \left\| {\rho (s)} \right\|_3^3ds + c{\mkern 1mu} \left\| \rho \right\|_{{L^\infty }(0, t;H)}^{3/2}\left\| \mu \right\|_{{L^2}(0, t;V)}^{3/2} \\ \leq \int_0^t {} {\left\| {\rho (s)} \right\|^3}ds + c{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^3{\mkern 1mu}. \\ \end{gathered}$

(2.45)

In addition, condition (2.5), Hälder’s inequality, and (2.42), yield that

$\begin{gathered} {J_3}:=-\int_{{Q_t}} {} (B[{\rho _1}]-B[{\rho _2}]){\mkern 1mu} \rho {\mkern 1mu} |\rho |dx ds \\ \leq c\int_0^t {} {\left\| {\rho (s)} \right\|_3}{\mkern 1mu} {\left\| {\rho (s)} \right\|_2}{\mkern 1mu} {\left\| {B[{\rho _1}](s)-B[{\rho _2}](s)} \right\|_6}ds \\ \leq c{\mkern 1mu} \mathop {\sup }\limits_{0 \leq s \leq t} {\left\| {\rho (s)} \right\|_3}{\mkern 1mu} {\left\| \rho \right\|_{{L^\infty }(0, t;H)}}\int_0^t {} {\left\| {\rho (s)} \right\|_H}ds \\ \leq \frac{1}{6}{\mkern 1mu} \mathop {\sup }\limits_{0 \leq s \leq t} \left\| {\rho (s)} \right\|_3^3 + c{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^3{\mkern 1mu}. \\ \end{gathered}$

(2.46)

Combining the estimates (2.43)-(2.46), and invoking Gronwall’s lemma, we can easily infer that

${\left\| \rho \right\|_{{L^\infty }(0, t;{L^3}(\Omega))}} \leq c{\mkern 1mu} {\left\| u \right\|_{{L^2}(0, t;H)}}\quad {\text{for all }}{\mkern 1mu} t \in (0, T].$

(2.47)

Step 3: With the above estimates proved, the road is paved for multiplying (2.33) by $\mu_t$. Integration over Q_t yields that

$\int_{{Q_t}} {} (1 + 2g({\rho _1})){\mkern 1mu} \mu _t^2dx ds + \frac{1}{2}{\mkern 1mu} \left\| {\nabla \mu (t)} \right\|_H^2 \leq \sum\limits_{j=1}^5 {\mkern 1mu} |{K_j}|,$

(2.48)

where the expressions K_j, 1≤j≤5, are estimated as follows: at first, using the global bounds and Young’s inequality, we have for every $\gamma>0$ (to be specified later) the bound

$\begin{gathered} {K_1}:={\mkern 1mu}-\int_{{Q_t}} {} g'({\rho _1}){\partial _t}{\rho _1}{\mkern 1mu} \mu {\mkern 1mu} {\mu _t}dx ds \leq \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }\int_{{Q_t}} {} {\mu ^2}dx ds \\ \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} \mu _t^2dx ds{\mkern 1mu} + \frac{c}{\gamma }\left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu}. \\ \end{gathered}$

(2.49)

Next, thanks to the mean value theorem, and employing (2.31) and (2.47), we find that

$\begin{gathered} {K_2}:={\mkern 1mu}-{\mkern 1mu} 2\int_{{Q_t}} {} (g({\rho _1})-g({\rho _2})){\mkern 1mu} {\partial _t}{\mu _2}{\mkern 1mu} {\mu _t}dx ds \leq c\int_{{Q_t}} {} |\rho |{\mkern 1mu} |{\partial _t}{\mu _2}|{\mkern 1mu} |{\mu _t}|dx ds \\ \leq c\int_0^t {} {\left\| {\rho (s)} \right\|_3}{\mkern 1mu} {\left\| {{\partial _t}{\mu _2}(s)} \right\|_6}{\mkern 1mu} {\left\| {{\mu _t}(s)} \right\|_2}ds \\ \leq \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }{\mkern 1mu} \left\| \rho \right\|_{{L^\infty }(0, t;{L^3}(\Omega))}^2\int_0^t {} \left\| {{\partial _t}{\mu _2}(s)} \right\|_V^2ds \\ \leq \gamma \int_{{Q_t}} {} \mu _t^2ds + \frac{c}{\gamma }{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu}. \\ \end{gathered}$

(2.50)

Moreover, we infer that

${K_3}:=\int_{{Q_t}} {} u{\mkern 1mu} {\mu _t}dx ds{\mkern 1mu} \leq \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu},$

(2.51)

as well as, invoking the mean value theorem once more,

$\begin{gathered} {K_4}:={\mkern 1mu}-\int_{{Q_t}} {} (g'({\rho _1})-g'({\rho _2})){\partial _t}{\rho _1}{\mkern 1mu} {\mu _2}{\mkern 1mu} {\mu _t}dx ds \leq c\int_{{Q_t}} {} |\rho |{\mkern 1mu} |{\mu _t}|dx ds \\ \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }\left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu}, \\ \end{gathered}$

(2.52)

and, finally, using (2.42) and Young’s inequality,

$\begin{gathered} {K_5}:={\mkern 1mu}-\int_{{Q_t}} {} g'({\rho _2}){\mkern 1mu} {\mu _2}{\mkern 1mu} {\rho _t}{\mkern 1mu} {\mu _t}dx ds \leq c\int_{{Q_t}} {} |{\rho _t}|{\mkern 1mu} |{\mu _t}|dx ds \\ \leq \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }{\mkern 1mu} \left\| \rho \right\|_{{H^1}(0, t;H)}^2 \\ \leq \gamma \int_{{Q_t}} {} \mu _t^2dx ds + \frac{c}{\gamma }{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu}. \\ \end{gathered}$

(2.53)

Now we combine the estimates (2.48)-(2.53) and choose > 0 appropriately small. It then follows that

${\mu _{{H^1}(0, t;H) \cap {L^\infty }(0, t;V)}} \leq c{\mkern 1mu} {\left\| u \right\|_{{L^2}(0, t;H)}}{\mkern 1mu}.$

(2.54)

Finally, we come back to (2.33) and employ the global bounds (2.16), (2.17), (2.31), and the estimates shown above, to conclude that

$\begin{gathered} {\left\| {\Delta \mu } \right\|_{{L^2}(0, t;H)}}{\mkern 1mu} \leq c\left({{{\left\| {{\mu _t}} \right\|}_{{L^2}(0, t;H)}}{\mkern 1mu} + {{\left\| \mu \right\|}_{{L^2}(0, t;H)}}{\mkern 1mu} + {{\left\| {{\rho _t}} \right\|}_{{L^2}(0, t;H)}}} \right. \\ \left. { + {{\left\| \rho \right\|}_{{L^2}(0, t;H)}}{\mkern 1mu} + {{\left\| u \right\|}_{{L^2}(0, t;H)}}} \right) + c{\left\| {\rho {\mkern 1mu} {\partial _t}{\mu _2}} \right\|_{{L^2}(0, t;H)}} \\ \leq c{\left\| u \right\|_{{L^2}(0, t;H)}}{\mkern 1mu}, \\ \end{gathered}$

(2.55)

where the last summand on the right-hand side was estimated as follows:

$\int_{{Q_t}} {} {\mkern 1mu} |\rho {|^2}{\mkern 1mu} |{\partial _t}{\mu _2}{|^2}dx ds \leq c\int_0^t {} \left\| {{\partial _t}{\mu _2}(s)} \right\|_6^2{\mkern 1mu} \left\| {\rho (s)} \right\|_3^2ds \leq c\left\| \rho \right\|_{{L^\infty }(0, t;{L^3}(\Omega))}^2\int_0^t {} \left\| {{\partial _t}{\mu _2}(s)} \right\|_V^2ds \leq c{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^2{\mkern 1mu}.$

Invoking standard elliptic estimates, we have thus shown that

${\left\| \mu \right\|_{{L^2}(0, t;W)}} \leq c{\left\| u \right\|_{{L^2}(0, t;H)}}{\mkern 1mu}.$

(2.56)

S_TEP 4: It remains to show the L^∞(0, t; L⁶(Ω))-estimate for $\rho$. To this end, we multiply (2.34) by $\rho|\rho|^4\,$ and integrate over Q_t. It follows that

$\frac{1}{6}\left\| {\rho (t)} \right\|_6^6{\mkern 1mu} \leq {\mkern 1mu} \sum\limits_{j=1}^3 | {L_j}|,$

(2.57)

where quantities L_j, 1≤j≤3, are estimated as follows: at first, we obtain from the global estimates (2.17) and (2.31), that

${L_1}:=\int_{{Q_t}} {} (-F'({\rho _1}) + F'({\rho _2}) + {\mu _2}(g'({\rho _1})-g'({\rho _2}))){\mkern 1mu} \rho {\mkern 1mu} |\rho {|^4}dx ds \leq c\int_0^t {} \left\| {\rho (s)} \right\|_6^6ds{\mkern 1mu}.$

(2.58)

Moreover, from (2.54) and Hälder’s and Young’s inequalities we conclude that

$\begin{gathered} {L_2}:=\int_{{Q_t}} {} g'({\rho _1}){\mkern 1mu} \mu {\mkern 1mu} \rho {\mkern 1mu} |\rho {|^4}dx ds \leq c\int_0^t {} {\left\| {\mu (s)} \right\|_6}\left\| {\rho (s)} \right\|_6^5ds \\ \leq c{\mkern 1mu} {\left\| \mu \right\|_{{L^\infty }(0, t;V)}}\left\| \rho \right\|_{{L^5}(0, t;{L^6}(\Omega))}^5 \leq c\left\| \mu \right\|_{{L^\infty }(0, t;V)}^6 + c{\mkern 1mu} \left\| \rho \right\|_{{L^5}(0, t;{L^6}(\Omega))}^6 \\ \leq c{\mkern 1mu} \left\| u \right\|_{{L^2}(0, t;H)}^6 + c\int_0^t {} \left\| {\rho (s)} \right\|_6^6ds{\mkern 1mu}. \\ \end{gathered}$

(2.59)

Finally, we employ (2.5) and (2.42) to infer that

$\begin{gathered} {L_3}:={\mkern 1mu}-\int_{{Q_t}} {} (B[{\rho _1}]-B[{\rho _2}]){\mkern 1mu} \rho {\mkern 1mu} |\rho {|^4}dx ds \leq c\int_0^t {} {\left\| {B({\rho _1}](s)-B[{\rho _2}](s)} \right\|_6}{\mkern 1mu} \left\| {\rho (s)} \right\|_6^5ds \\ \leq c{\mkern 1mu} \mathop {\sup }\limits_{0 \leq s \leq t} \left\| {\rho (s)} \right\|_6^5\int_0^t {} {\left\| {\rho (s)} \right\|_H}ds \leq \frac{1}{{12}}{\mkern 1mu} \mathop {\sup }\limits_{0 \leq s \leq t} \left\| {\rho (s)} \right\|_6^6{\mkern 1mu} + c\left\| u \right\|_{{L^2}(0, t;H)}^6{\mkern 1mu}. \\ \end{gathered}$

(2.60)

Combining the estimates (2.57)-(2.60), and invoking Gronwall’s lemma, then we readily find the estimate

${\left\| \rho \right\|_{{L^\infty }(0, t;{L^6}(\Omega))}} \leq c{\mkern 1mu} {\left\| u \right\|_{{L^2}(0, t;H)}}{\mkern 1mu},$

which concludes the proof of the assertion.

---

3. Directional differentiability of the control-to-state mapping

In this section, we prove the relevant differentiability properties of the solution operator S. To this end, we introduce the spaces

$\begin{gathered} X:={H^1}(0, T;H) \cap {L^\infty }(Q), \hfill \\ Y:={H^1}(0, T;H) \times \left({{L^\infty }(0, T;H) \cap {L^2}(0, T;V)} \right), \hfill \\ \end{gathered}$

endowed with their natural norms

$\begin{gathered} {\left\| u \right\|_X}:={\mkern 1mu} {\left\| u \right\|_{{H^1}(0, T;H)}}{\mkern 1mu} + {\left\| u \right\|_{{L^\infty }(Q)}}\quad \forall {\mkern 1mu} u \in X, \hfill \\ {\left\| {(\rho, \mu)} \right\|_Y}:={\left\| \rho \right\|_{{H^1}(0, T;H)}}{\mkern 1mu} + {\left\| \mu \right\|_{{L^\infty }(0, T;H)}}{\mkern 1mu} + {\left\| \mu \right\|_{{L^2}(0, T;V)}}\quad \forall {\mkern 1mu} (\rho, \mu) \in Y{\mkern 1mu}, \hfill \\ \end{gathered}$

and consider the control-to-state operator S as a mapping between ${U_{{\text{ad}}}} \subset X$ and Y. Now let $\bar u \in {U_{{\text{ad}}}}$ be fixed and put $(\bar \rho, \bar u):=S(\bar u)$. We then study the linearization of the state system (1.2)-(1.5) at $\bar u$, which is given by:

$\begin{gathered} (1 + 2g(\bar \rho)){\mkern 1mu} {\eta _t}{\mkern 1mu} + {\mkern 1mu} 2g'(\bar \rho){{\bar u}_t}{\mkern 1mu} \xi {\mkern 1mu} + {\mkern 1mu} g'(\bar \rho){{\bar \rho }_t}{\mkern 1mu} \eta {\mkern 1mu} + {\mkern 1mu} \bar u{\mkern 1mu} g''(\bar \rho){\mkern 1mu} {{\bar \rho }_t}{\mkern 1mu} \xi {\mkern 1mu} + {\mkern 1mu} \bar ug'(\bar \rho){\mkern 1mu} {\xi _t} \hfill \\-\Delta \eta=h\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in }}Q, \hfill \\ \end{gathered}$

(3.1)

${\xi _t} + F''(\bar \rho){\mkern 1mu} \xi + DB[\bar \rho](\xi)=\bar ug''(\bar \rho){\mkern 1mu} \xi {\mkern 1mu} + {\mkern 1mu} g'(\bar \rho){\mkern 1mu} \eta \quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in }}{\mkern 1mu} Q,$

(3.2)

${\partial _{\text{n}}}\eta=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{on }}{\mkern 1mu} \Sigma,$

(3.3)

$\eta (0)=\xi (0)=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in}}{\mkern 1mu} {\text{ }}\Omega.$

(3.4)

Here, $h \in X$ must satisfy $\bar u + \bar \lambda h \in {U_{{\text{ad}}}}$ for some $\bar \lambda > 0$. Provided that the system (3.1)-(3.4) has for any such h a unique solution pair $(\xi, \eta)$, we expect that the directional derivative $\delta S(\bar u; h)$ of S at $\bar u$ in the direction h (if it exists) ought to be given by $(\xi, \eta)$. In fact, the above problem makes sense for every $h \in {L^2}(Q)$, and it is uniquely solvable under this weaker assumption.

T_HEOREM 3.1: Suppose that the general hypotheses (A1)-(A5) are satisfied and let $h \in {L^2}(Q)$. Then, the linearized problem (3.1)-(3.4) has a unique solution $(\xi, \eta)$ satisfying

$\xi \in {H^1}(0, T;H) \cap {L^\infty }(0, T;{L^6}(\Omega)),$

(3.5)

$\eta \in {H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^2}(0, T;W).$

(3.6)

P_ROOF: We first prove uniqueness. Since the problem is linear, we take h=0 and show that $(\xi, \eta)$=(0, 0). We add $\eta$ and $\xi$ to both sides of equations (3.1) and (3.2), respectively, then multiply by $\eta$ and $\xi_t$, integrate over Q_t, and sum up. By observing that

$(1 + 2g(\bar \rho)){\eta _t}\eta + g'(\bar \rho){{\bar \rho }_t}|\eta {|^2}={\partial _t}[(\frac{1}{2} + g(\bar \rho))|\eta {|^2}],$

and recalling that g≥0, we obtain

$\frac{1}{2}\int_\Omega {} |\eta (t){|^2}dx + \int_0^t {} \left\| {\eta (s)} \right\|_V^2ds + \frac{1}{2}{\text{ }}\int_\Omega {} |\xi (t){|^2}dx + \int_{{Q_t}} {} |{\xi _t}{|^2}dx ds \leq \sum\limits_{j=1}^3 {{H_j}},$

where the terms H_j are defined and estimated as follows. We have

$\begin{gathered} {H_1}:=-\int_{{Q_t}} {} 2{\mkern 1mu} g'(\bar \rho){\mkern 1mu} {{\bar u}_t}{\mkern 1mu} \xi {\mkern 1mu} \eta dx ds \leq c\int_0^t {} {\left\| {{{\bar u}_t}(s)} \right\|_3}{\left\| {\xi (s)} \right\|_2}{\left\| {\eta (s)} \right\|_6}ds \hfill \\ \leq \frac{1}{2}\int_0^t {} \left\| {\eta (s)} \right\|_V^2ds + c\int_0^t {} \left\| {{{\bar u}_t}(s)} \right\|_V^2\left\| {\xi (s)} \right\|_2^2ds{\mkern 1mu}, \hfill \\ \end{gathered}$

and we notice that the function $s \mapsto \left\| {{{\bar u}_t}(s)} \right\|_V^2$ belongs to L¹(0, T), by (2.30) for $\bar u$. Next, we easily have the estimate

$\begin{gathered} {H_2}:=\int_{{Q_t}} {} (\eta-\bar u{\mkern 1mu} g''(\bar \rho){\mkern 1mu} {{\bar \rho }_t}{\mkern 1mu} \xi-\bar ug'(\bar \rho){\mkern 1mu} {\xi _t}){\mkern 1mu} \eta dx ds \hfill \\ \leq \frac{1}{4}\int_{{Q_t}} {} |{\xi _t}{|^2}dx ds + c\int_{{Q_t}} {} (|\xi {|^2} + |\eta {|^2})dx ds. \hfill \\ \end{gathered}$

Finally, recalling (2.8), it is clear that

$\begin{gathered} {H_3}:=\int_{{Q_t}} {} ((\xi + \bar ug''(\bar \rho)-F''(\bar \rho)){\mkern 1mu} \xi-DB[\bar \rho](\xi) + g'(\bar \rho){\mkern 1mu} \eta){\mkern 1mu} {\xi _t}dx ds \hfill \\ \leq \frac{1}{4}\int_{{Q_t}} {} |{\xi _t}{|^2}dx ds + c\int_{{Q_t}} {} (|\xi {|^2} + |\eta {|^2})dx ds. \hfill \\ \end{gathered}$

(3.7)

Therefore, it suffices to collect these inequalities and apply Gronwall’s lemma in order to conclude that $\xi$=0 and $\eta$=0.

The existence of a solution is proved in several steps. First, we introduce an approximating problem depending on the parameter $\varepsilon \in (0, 1)$. Then, we show well-posedness for this problem and perform suitable a priori estimates. Finally, we construct a solution to problem (3.1)-(3.4) by letting $\varepsilon$ tend to zero. For the sake of simplicity, in performing the uniform a priori estimates, we denote by c > 0 different constants that may depend on the data of the system but not on $\varepsilon \in (0, 1)$; the actual value of c may change within formulas and lines. On the contrary, the symbol $c_\varepsilon$ stands for (different) constants that can depend also on $\varepsilon$. In particular, $c_\varepsilon$ is independent of the parameter $\delta$ that enters an auxiliary problem we introduce later on.

S_TEP 1: We approximate $\bar \rho$ and $\bar \mu$ by suitable ${\rho ^\varepsilon }, {\mu ^\varepsilon } \in {C^\infty }(\bar Q)$ as specified below. For every $\varepsilon \in (0, 1)$, it holds that

${\rho _{**}} \leq {\rho ^\varepsilon } \leq {\rho ^{**}}{\text{ in }}\bar Q {\text{ and }}{\left\| {\rho _t^\varepsilon } \right\|_{{L^\infty }(Q)}} + {\left\| {{\mu ^\varepsilon }} \right\|_{{H^1}(0, T;{L^3}(\Omega)) \cap {L^\infty }(Q)}} \leq {C^*},$

(3.8)

for some constants $\rho_{**}, \rho^{**}\in(0, 1)$ and $C^*>0$; as $\varepsilon \searrow 0$, we have

$\begin{gathered} {\rho ^\varepsilon } \to \bar \rho, {\rho ^\varepsilon }_t \to {{\bar \rho }_t}, {\mu ^\varepsilon } \to \bar \mu, \;{\text{in }}{L^p}(Q), {\text{ for every }}p < + \infty {\text{ and a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in }}Q, \hfill \\ and\mu _t^\varepsilon \to {{\bar \mu }_t}\; \;{\text{in }}{L^2}(0, T;{L^3}(\Omega)). \hfill \\ \end{gathered}$

(3.9)

In order to construct regularizing families as above, we can use, for instance, extension outside Q and convolution with mollifiers.

Next, we introduce the approximating problem of finding (${\xi ^\varepsilon }, {\eta ^\varepsilon }$) satisfying

$\xi _t^\varepsilon + F''(\bar \rho){\xi ^\varepsilon } + DB[\bar \rho]({\xi ^\varepsilon })=\bar \mu g''(\bar \rho){\xi ^\varepsilon } + g'(\bar \rho){\eta ^\varepsilon }\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in}}{\mkern 1mu} {\text{ }}Q,$

(3.10)

$\begin{gathered} (1 + 2g({\rho ^\varepsilon }))\eta _t^\varepsilon + g'({\rho ^\varepsilon })\rho _t^\varepsilon {\eta ^\varepsilon } \hfill \\ + {\mkern 1mu} 2g'(\bar \rho)\mu _t^\varepsilon {\mkern 1mu} {\xi ^\varepsilon } + \bar \mu g''(\bar \rho){{\bar \rho }_t}{\xi ^\varepsilon } + \bar \mu g'(\bar \rho){\mkern 1mu} \xi _t^\varepsilon-\Delta {\eta ^\varepsilon }=h\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in}}{\mkern 1mu} {\text{ }}Q, \hfill \\ \end{gathered}$

(3.11)

${\partial _{\text{n}}}{\eta ^\varepsilon }=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{on }}{\mkern 1mu} \Sigma,$

(3.12)

${\eta ^\varepsilon }(0)={\xi ^\varepsilon }(0)=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in }}\Omega.$

(3.13)

In order to solve (3.10)-(3.13), we introduce the spaces

$V:=H \times V{\text{ and }}H:=H \times H,$

and present our problem in the form

$\frac{d}{{dt}}{\mkern 1mu} (\xi, \eta) + A(\xi, \eta)=f{\text{ and }}(\xi, \eta)(0)=(0, 0),$

in the framework of the Hilbert triplet (V, H, V′). We look for a weak solution and aim at applying [1, Thm. 3.2, p. 256]. To this end, we have to split ${A^\varepsilon }$ in the form ${Q^\varepsilon }+ {R^\varepsilon }$, where Q" is the uniformly elliptic principal part and the remainder ${R^\varepsilon }$ satisfies the requirements [1, (4.4)-(4.5), p. 259]. We notice at once that these conditions are trivially fulfilled whenever

${R^\varepsilon }=(R_1^\varepsilon, R_2^\varepsilon) \in L({L^2}(0, T;H), {L^2}(0, T;H)){\mkern 1mu},$

(3.14)

$\left| {\int_{{Q_t}} {} (R_1^\varepsilon (v, w){\mkern 1mu} v + R_2^\varepsilon (v, w){\mkern 1mu} w)dx ds} \right| \leq {C_{{R^\varepsilon }}}\int_{{Q_t}} {} (|v{|^2} + |w{|^2})ds{\mkern 1mu},$

(3.15)

for some constant ${C_{{R^\varepsilon }}}$, and every $v, w \in {L^2}(0, T;H)$ and $t\in[0, T]$. In order to present (3.10)-(3.13) in the desired form, we multiply (3.11) by ${a^\varepsilon }:=1/(1 + 2g({\rho ^\varepsilon }))$ and notice that

$-{a^\varepsilon }\Delta {\eta ^\varepsilon }=-{\text{div}}({a^\varepsilon }\Delta {\eta ^\varepsilon }) + \nabla {a^\varepsilon } \cdot \nabla {\eta ^\varepsilon }.$

As ${a^\varepsilon } \geq \alpha :=1/(1 + 2\sup g)$ and $\nabla {a^\varepsilon }$ is bounded, we can fix a real number ${\lambda ^\varepsilon } > 0$ such that

$\int_\Omega {} {a^\varepsilon }(t)|\nabla w{|^2} + (\nabla {a^\varepsilon }(t) \cdot \nabla w)w + {\lambda ^\varepsilon }|w{|^2})dx \geq \frac{\alpha }{2}\left\| w \right\|_V^2$

(3.16)

for every $w\in V$ and $t\in[0, T]$. Next, we replace $\xi _t^\varepsilon$ in (3.11) by using (3.10). Therefore, we see that a possible weak formulation of (3.10)-(3.12) is given by

$\begin{gathered} \int_\Omega {} \xi _t^\varepsilon (t)v dx{ + _{V'}}{\left\langle {\eta _t^\varepsilon (t), w} \right\rangle _V}{ + _{V'}}{\left\langle {{Q^\varepsilon }(t)({\xi ^\varepsilon }, {\eta ^\varepsilon })(t), (v, w)} \right\rangle _V} \hfill \\ + \int_\Omega {} (R_1^\varepsilon ({\xi ^\varepsilon }, {\eta ^\varepsilon })(t)v + R_2^\varepsilon ({\xi ^\varepsilon }, {\eta ^\varepsilon })(t){\mkern 1mu} w)dx=\int_\Omega {} {a^\varepsilon }(t){\mkern 1mu} h(t){\mkern 1mu} w dx \hfill \\ {\text{for a}}{\text{.}}\; {\text{a}}{\text{. }}t \in (0, T){\text{ and every }}(v, w) \in V, \hfill \\ \end{gathered}$

(3.17)

where the symbols $\left\langle { \cdot, \cdot } \right\rangle$ stand for the duality pairings and ${Q^\varepsilon }$ and $R_i^\varepsilon$ have the meaning explained below. The time-dependent operator ${Q^\varepsilon }$(t) from V into V′ is defined by

$\begin{gathered} _{{V^\prime }}{\left\langle {{Q^\varepsilon }(t)(\hat v, \hat w), (v, w)} \right\rangle _V} \hfill \\=\int_\Omega {} (\hat v{\mkern 1mu} v + {a^\varepsilon }(t){\mkern 1mu} \nabla \hat w \cdot \nabla w + (\nabla {a^\varepsilon }(t) \cdot \nabla \hat w){\mkern 1mu} w + {\lambda ^\varepsilon }{\mkern 1mu} \hat w{\mkern 1mu} w)dx \hfill \\ \end{gathered}$

(3.18)

for every $(\hat v, \hat w), (v, w) \in V$ and $t\in[0, T]$. By construction, the bilinear form given by the right-hand side of (3.18) is continuous on $V \times V$, depends smoothly on time, and is V-coercive uniformly with respect to t (see (3.16)). The operators

$R_i^\varepsilon \in L({L^2}(0, T;H), {L^2}(0, T;H))$

account for the term ${\lambda ^\varepsilon }{\eta ^\varepsilon }$ that has to be added also to the right-hand side of (3.11) and for all the contributions to the equations that have not been considered in the principal part. They have the form

$(R_i^\varepsilon (v, w))(t)=a_{i1}^\varepsilon (t)v + a_{i2}^\varepsilon (t)w + a_{i3}^\varepsilon (t)(DB[\bar \rho](v))(t)$

(3.19)

for $(v, w) \in {L^2}(0, T;H)$, with some coefficients $a_{ij}^\varepsilon \in {L^\infty }(Q)$. Therefore, by virtue of (2.8), we see that

$\begin{gathered} \int_{{Q_t}} {} (R_1^\varepsilon (v, w){\mkern 1mu} v + R_1^\varepsilon (v, w){\mkern 1mu} w)dx ds \hfill \\ \leq c\int_{{Q_t}} {} (|v{|^2} + |w{|^2})dx ds + c\left\| {DB[\bar \rho](v)} \right\|_{{L^2}({Q_t})}^2 \hfill \\ \leq c\int_{{Q_t}} {} (|v{|^2} + |w{|^2})dx ds{\mkern 1mu}, \hfill \\ \end{gathered}$

for every $(v, w) \in {L^2}(0, T;H)$ and every $t\in[0, T]$. Thus, the conditions (3.14)-(3.15) are fulfilled, and the result of ^[1] mentioned above can be applied. We conclude that the Cauchy problem for (3.17) has a unique solution (${\xi ^\varepsilon }, {\eta ^\varepsilon }$) satisfying

$\begin{gathered} ({\xi ^\varepsilon }, {\eta ^\varepsilon }) \in {H^1}(0, T;V') \cap {L^2}(0, T;V), \quad {\text{i}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{., }} \hfill \\ {\xi ^\varepsilon } \in {H^1}(0, T;H)and{\eta ^\varepsilon } \in {H^1}(0, T;V') \cap {L^2}(0, T;V). \hfill \\ \end{gathered}$

On the other hand, this solution has to satisfy

$\left\langle {{\partial _t}{\eta ^\varepsilon }, w} \right\rangle + \int_\Omega {} {a^\varepsilon }\nabla {\eta ^\varepsilon } \cdot \nabla w dx=\int_\Omega {} {\phi _\varepsilon }{\mkern 1mu} w{\text{ }}dx\quad {\text{ a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in (0, T), for}} {\text{every }}w \in V,$

for some ${\phi _\varepsilon } \in {L^2}(\Omega)$. From standard elliptic regularity, it follows that ${\eta ^\varepsilon } \in {H^1}(0, T;H) \cap {L^2}(0, T;W)$.

In the next steps, besides of Young’s inequality, we make repeated use of the global estimates (2.16), (2.17), and (2.31), for ${\bar \rho }$ and ${\bar \mu }$, without further reference.

S_TEP 2: For convenience, we refer to Eqs. (3.10)-(3.12) (using the language that is proper for strong solutions), but it is understood that they are meant in the variational sense (3.17). We add ${\xi ^\varepsilon }$ and ${\eta ^\varepsilon }$ to both sides of (3.10) and (3.11), respectively; then, we multiply the resulting equalities by $\xi _t^\varepsilon$ and ${\eta ^\varepsilon }$, integrate over Q_t, and sum up. By observing that

$(1 + 2g({\rho ^\varepsilon })){\mkern 1mu} \eta _t^\varepsilon {\eta ^\varepsilon } + g'({\rho ^\varepsilon }){\mkern 1mu} \rho _t^\varepsilon |{\eta ^\varepsilon }{|^2}={\partial _t}[(\frac{1}{2} + g({\rho ^\varepsilon })){\mkern 1mu} |{\eta ^\varepsilon }{|^2}],$

and recalling that g≥0, we obtain

$\frac{1}{2}\int_\Omega {} |{\xi ^\varepsilon }(t){|^2}dx + \int_{{Q_t}} {} |\xi _t^\varepsilon {|^2}dx ds + \frac{1}{2}\int_\Omega {} |{\eta ^\varepsilon }(t){|^2}dx + \int_0^t {} \left\| {{\eta ^\varepsilon }(s)} \right\|_V^2ds \leq \sum\limits_{j=1}^3 {{I_j}},$

where the terms I_j are defined and estimated as follows. In view of (2.8), we first infer that

$\begin{gathered} {I_1}:=\int_{{Q_t}} {} ({\xi ^\varepsilon }-F''(\bar \rho){\mkern 1mu} {\xi ^\varepsilon }-DB[\bar \rho]({\xi ^\varepsilon }) + \bar \mu g''(\bar \rho){\mkern 1mu} {\xi ^\varepsilon } + g'(\bar \rho){\mkern 1mu} {\eta ^\varepsilon })\xi _t^\varepsilon dx ds \hfill \\ \leq \frac{1}{4}\int_{{Q_t}} {} |\xi _t^\varepsilon {|^2}dx ds + c\int_{{Q_t}} {} (|{\xi ^\varepsilon }{|^2} + |{\eta ^\varepsilon }{|^2})dx ds. \hfill \\ \end{gathered}$

Next, we have

$\begin{gathered} {I_2}:=\int_{{Q_t}} {} ({\eta ^\varepsilon }-\bar \mu g''(\bar \rho){\mkern 1mu} {{\bar \rho }_t}{\mkern 1mu} {\xi ^\varepsilon }-\bar \mu g'(\bar \rho)\xi _t^\varepsilon + h){\eta ^\varepsilon }dx ds \hfill \\ \leq \frac{1}{4}\int_{{Q_t}} {} |\xi _t^\varepsilon {|^2}dx ds + c\int_{{Q_t}} {} (|{\xi ^\varepsilon }{|^2} + |{\eta ^\varepsilon }{|^2})dx ds + c{\mkern 1mu}. \hfill \\ \end{gathered}$

Finally, by virtue of the H¨older and Sobolev inequalities, we have

$\begin{gathered} {I_3}:=-\int_{{Q_t}} {} 2g'(\bar \rho)\mu _t^\varepsilon {\xi ^\varepsilon }{\eta ^\varepsilon }dx ds \leq c\int_0^t {} {\left\| {\mu _t^\varepsilon (s)} \right\|_3}{\left\| {{\mkern 1mu} {\xi ^\varepsilon }(s)} \right\|_2}{\mkern 1mu} {\left\| {{\eta ^\varepsilon }(s)} \right\|_6}ds \hfill \\ \leq \frac{1}{2}\int_0^t {} \left\| {{\eta ^\varepsilon }(s)} \right\|_V^2ds + c\int_0^t {} \left\| {{\eta ^\varepsilon }(s)} \right\|_3^2\left\| {{\mkern 1mu} {\xi ^\varepsilon }(s)} \right\|_2^2ds{\mkern 1mu}. \hfill \\ \end{gathered}$

At this point, we recall all the inequalities we have proved, notice that (3.8) implies that the function $s \mapsto \left\| {\mu _t^\varepsilon (s)} \right\|_3^2$ is bounded in L¹(0, T), and apply the Gronwall lemma. We obtain

${\left\| {{\xi ^\varepsilon }} \right\|_{{H^1}(0, T;H)}} + {\left\| {{\eta ^\varepsilon }} \right\|_{{L^\infty }(0, T;H) \cap {L^2}(0, T;V)}} \leq c{\mkern 1mu}.$

(3.20)

S_TEP 3: We would now like to test (3.10) by (${\xi ^\varepsilon }$)⁵. However, this function is not admissible, unfortunately. Therefore, we introduce a suitable approximation. To start with, we consider the Cauchy problem

${{\hat \xi }_t} + b{\mkern 1mu} \hat \xi + L(\hat \xi)={f^\varepsilon }{\text{ and }}\hat \xi (0)=0,$

(3.21)

where we have set, for brevity,

$b:=F''(\bar \rho)-\bar \mu {\mkern 1mu} g''(\bar \rho), \quad L:=DB[\bar \rho], {\text{ and }}{f^\varepsilon }:=g'(\bar \rho){\mkern 1mu} {\eta ^\varepsilon }.$

(3.22)

By (3.10), $\hat \xi :={\xi ^\varepsilon }$ is a solution belonging to H¹(0, T; H). On the other hand, such a solution is unique. Indeed, multiplying by $\hat \xi$ the corresponding homogeneous equation (i. e., ${f^\varepsilon }$ is replaced by 0), and invoking (2.8) and Gronwall’s lemma, one immediately obtains that $\hat \xi$=0. We conclude that $\hat \xi :={\xi ^\varepsilon }$ is the unique solution to (3.21).

At this point, we approximate ${\xi ^\varepsilon }$ by the solution to a problem depending on the parameter $\delta\in(0, 1)$, in addition. Namely, we look for ${\xi ^{\varepsilon \delta }}$ satisfying the parabolic-like equation

$\xi _t^{\varepsilon \delta }-\delta {\mkern 1mu} \Delta {\xi ^{\varepsilon \delta }} + {b^\delta }{\xi ^{\varepsilon \delta }} + L({\xi ^{\varepsilon \delta }})={f^\varepsilon },$

(3.23)

complemented with the Neumann boundary condition ${\partial _{\text{n}}}{\xi ^{\varepsilon \delta }}=0$ and the initial condition $${\xi ^{\varepsilon \delta }}=0$$ . In (3.23), ${b^\delta }$ is an approximation of b belonging to $C^\infty(\overline Q)$ that satisfies

${\left\| {{b^\delta }} \right\|_{{L^\infty }(Q)}} \leq c, {\text{ and }}{b^\delta } \to b\quad {\text{ a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{in }}Q{\text{ as }}\delta \searrow 0.$

(3.24)

This problem has a unique weak solution ${\xi ^{\varepsilon \delta }} \in {H^1}(0, T;V') \cap {L^2}(0, T;V)$, as one easily sees by arguing as we did for the more complicated system (3.10)-(3.13) and applying [1, Thm. 3.2, p. 256].

We now aim to show that ${\xi ^{\varepsilon \delta }}$ is bounded. To this end, we introduce the operator ${A_\delta } \in L(V, V')$ defined by

$\left\langle {{A_\delta }v, w} \right\rangle :=\int_\Omega {} (\delta {\mkern 1mu} \nabla v \cdot \nabla w + v{\mkern 1mu} w)dx\quad {\text{for every }}v, w \in V,$

and observe that ${A_\delta }$ is an isomorphism. Moreover, Eq. (3.23), complemented with the boundary and initial conditions, can be written as

$\xi _t^{\varepsilon \delta } + {A_\delta }{\xi ^{\varepsilon \delta }}={f^{\varepsilon \delta }}:={f^\varepsilon }-(1 + {b^\delta }){\xi ^{\varepsilon \delta }} + L({\xi ^{\varepsilon \delta }}) and {\xi ^{\varepsilon \delta }}(0)=0$

(3.25)

Now, by also accounting for (2.9), we notice that ${f^\varepsilon }, {\xi ^{\varepsilon \delta }}, {b^\delta }{\xi ^{\varepsilon \delta }}$, and $L({\xi ^{\varepsilon \delta }})$, all belong to L²(0, T; V). Hence, ${f^{\varepsilon \delta }} \in {L^2}(0, T;V)$, so that ${A_\delta }{f^{\varepsilon \delta }} \in {L^2}(0, T;V')$, and we can consider the unique solution ${\zeta ^{\varepsilon \delta }} \in {H^1}(0, T;V') \cap {L^2}(0, T;V)$ to the problem

$\zeta _t^{\varepsilon \delta } + {A_\delta }{\zeta ^{\varepsilon \delta }}={A_\delta }{f^{\varepsilon \delta }}{\text{ and }}{\zeta ^{\varepsilon \delta }} (0)=0$

Now, $A_\delta ^{-1}{\zeta ^{\varepsilon \delta }}$ satisfies

${(A_\delta ^{-1}{\zeta ^{\varepsilon \delta }})_t} + {A_\delta }(A_\delta ^{-1}{\zeta ^{\varepsilon \delta }})=A_\delta ^{-1}{A_\delta }{f^{\varepsilon \delta }}={f^{\varepsilon \delta }}{\text{ and }}(A_\delta ^{-1}{\zeta ^{\varepsilon \delta }})(0)=0,$

so that a comparison with (3.25) shows that ${\xi ^{\varepsilon \delta }}=A_\delta ^{-1}{\zeta ^{\varepsilon \delta }}$, by uniqueness. Since ${\zeta ^{\varepsilon \delta }} \in {L^\infty }(0, T;H)$, and $A_\delta ^{-1}(H)=W$ by elliptic regularity, we deduce that ${\xi ^{\varepsilon \delta }} \in {L^\infty }(0, T;W)$. Therefore, ${\xi ^{\varepsilon \delta }}$ is bounded, as claimed.

Consequently, (${\xi ^{\varepsilon \delta }}$)⁵ is an admissible test function, since it clearly belongs to the space L²(0, T; V). By multiplying (3.23) by (${\xi ^{\varepsilon \delta }}$)⁵ and integrating over Q_t, we obtain that

$\frac{1}{6}\int_\Omega {} |{\xi ^{\varepsilon \delta }}(t){|^6}{\mkern 1mu} dx + {\mkern 1mu} 5{\mkern 1mu} \delta \int_{{Q_t}} {} |{\xi ^{\varepsilon \delta }}{|^4}{\mkern 1mu} |\nabla {\xi ^{\varepsilon \delta }}{|^2}dx ds=\sum\limits_{j=1}^3 {{K_j}},$

where the terms K_j are defined and estimated as follows. First, recalling (3.24), we deduce that

${K_1}:=-\int_{{Q_t}} {} {b^\delta }{\xi ^{\varepsilon \delta }}{\mkern 1mu} {({\xi ^{\varepsilon \delta }})^5}dx ds \leq c\int_{{Q_t}} {} |{\xi ^\varepsilon }{|^6}dx ds{\mkern 1mu}.$

On the other hand, Hälder’s inequality, and assumption (2.8) with p=6, imply that

$\begin{gathered} {K_2}:=-\int_{{Q_t}} {} L({\xi ^{\varepsilon \delta }}){\mkern 1mu} {({\xi ^{\varepsilon \delta }})^5}dx ds \leq c{\left\| {L{\xi ^{\varepsilon \delta }}} \right\|_{{L^6}({Q_t})}}{\left\| {{{({\xi ^{\varepsilon \delta }})}^5}} \right\|_{{L^{6/5}}({Q_t})}} \hfill \\ \leq c{\mkern 1mu} {\left\| {{\xi ^{\varepsilon \delta }}} \right\|_{{L^6}({Q_t})}}{\mkern 1mu} \left\| {{\xi ^{\varepsilon \delta }}} \right\|_{{L^6}({Q_t})}^5{\mkern 1mu}={\mkern 1mu} c\int_{{Q_t}} {} |{\xi ^{\varepsilon \delta }}{|^6}dx ds. \hfill \\ \end{gathered}$

Finally, also invoking Sobolev’s inequality, we see that

$\begin{gathered} {K_3}:=\int_{{Q_t}} {} {f^\varepsilon }{({\xi ^{\varepsilon \delta }})^5}dx ds \leq c\int_0^t {} {\left\| {{\eta ^\varepsilon }(s)} \right\|_6}{\mkern 1mu} {\left\| {{{({\xi ^{\varepsilon \delta }}(s))}^5}} \right\|_{6/5}}ds \hfill \\ \leq c\int_0^t {} {\left\| {{\eta ^\varepsilon }(s)} \right\|_6}\left\| {{\xi ^{\varepsilon \delta }}(s)} \right\|_6^5ds \leq c\int_0^t {} \left\| {{\eta ^\varepsilon }(s)} \right\|{{\mkern 1mu} _V}(1 + \left\| {{\xi ^{\varepsilon \delta }}(s)} \right\|_6^6)ds{\mkern 1mu}. \hfill \\ \end{gathered}$

Collecting the above estimates, and noting that the function $s \mapsto {\left\| {{\eta ^\varepsilon }(s)} \right\|_V}$ is bounded in L¹(0, T) by (3.20), we can apply the Gronwall lemma to conclude that

${\left\| {{\xi ^{\varepsilon \delta }}} \right\|_{{L^\infty }(0, T;{L^6}(\Omega))}} \leq c{\mkern 1mu}.$

(3.26)

At this point, we quickly show that ${\xi ^{\varepsilon \delta }}$ converges to ${\xi ^\varepsilon }$ as $\delta\searrow0$, at least for a subsequence. Indeed, one multiplies (3.23) first by ${\xi ^{\varepsilon \delta }}$, and then by $\xi _t^\varepsilon$, and proves that

${\left\| {{\xi ^{\varepsilon \delta }}} \right\|_{{H^1}(0, T;H) \cap {L^\infty }(0, T;V)}} \leq {c_\varepsilon },$

uniformly with respect to $\delta$. Then, by weak compactness and (3.24) (which implies convergence of $b^\delta$ to b in L^p(Q) for every p < +∞), it is straightforward to see that ${\xi ^{\varepsilon \delta }}$ converges to a solution $\hat\xi$ to the problem associated with (3.21). As $\hat \xi={\xi ^\varepsilon }$, we have proved what we have claimed. This, and (3.26), yield that

${\left\| {{\xi ^\varepsilon }} \right\|_{{L^\infty }(0, T;{L^6}(\Omega))}} \leq c.$

(3.27)

S_TEP 4: At this point, we can multiply (3.11) by $\eta _t^\varepsilon$ and integrate over Q_t. By recalling that g≥0, we obtain

$\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + \frac{1}{2}\int_\Omega {} |\nabla {\eta ^\varepsilon }(t){|^2}dx \leq \sum\limits_{j=1}^3 {{L_j}},$

where each term L_j is defined and estimated below. First, by taking advantage of (3.27) and (3.8) for $\mu _t^\varepsilon$, we have

$\begin{gathered} {L_1}:=-\int_{{Q_t}} {} 2g'(\bar \rho)\mu _t^\varepsilon {\xi ^\varepsilon }\eta _t^\varepsilon dx ds \leq c\int_0^t {} {\left\| {\mu _t^\varepsilon (s)} \right\|_3}{\left\| {{\xi ^\varepsilon }(s)} \right\|_6}{\mkern 1mu} {\left\| {\eta _t^\varepsilon (s)} \right\|_2}ds \\ \leq c\int_0^t {} {\left\| {\mu _t^\varepsilon (s)} \right\|_3}{\left\| {\eta _t^\varepsilon (s)} \right\|_2}ds \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c\int_0^T {} \left\| {\mu _t^\varepsilon (s)} \right\|_3^2ds \\ \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c{\mkern 1mu}. \\ \end{gathered}$

Next, using (3.8) for $\rho _t^\varepsilon$ and (3.20), we obtain that

${L_2}:=-\int_{{Q_t}} {} g'({\rho ^\varepsilon })\rho _t^\varepsilon {\eta ^\varepsilon }\eta _t^\varepsilon dx ds \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c\int_{{Q_t}} {} |{\eta ^\varepsilon }{|^2}dx ds \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c{\mkern 1mu}.$

Finally, in view of (3.20), we have

$\begin{gathered} {L_3}:=\int_{{Q_t}} {} (-\bar \mu g''(\bar \rho){{\bar \rho }_t}{\mkern 1mu} {\xi ^\varepsilon }-\bar \mu g'(\bar \rho)\xi _t^\varepsilon + h)\eta _t^\varepsilon dx ds \\ \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c\int_{{Q_t}} {} (|{\xi ^\varepsilon }{|^2} + |\xi _t^\varepsilon {|^2} + 1)dx ds \leq \frac{1}{4}\int_{{Q_t}} {} |\eta _t^\varepsilon {|^2}dx ds + c{\mkern 1mu}. \\ \end{gathered}$

By collecting the above estimates, we conclude that

${\left\| {\eta _t^\varepsilon } \right\|_{{L^2}(0, T;H)}} + {\left\| {{\eta ^\varepsilon }} \right\|_{{L^2}(0, T;V)}}{\mkern 1mu} \leq c{\mkern 1mu}.$

(3.28)

As a consequence, we can estimate $\Delta {\eta ^\varepsilon }$ in L²(Q), just by comparison in (3.11). Using standard elliptic regularity, we deduce that

${\left\| {{\eta ^\varepsilon }} \right\|_{{L^2}(0, T;W)}}{\mkern 1mu} \leq c{\mkern 1mu}.$

(3.29)

S_TEP 5: At this point, we are ready to prove the existence part of the statement. Indeed, the estimates (3.20) and (3.27)-(3.29) yield that

$\begin{gathered} {\xi ^\varepsilon } \to \xi \quad {\text{weakly}} {\text{star}} {\text{in}} {H^1}(0, T;H) \cap {L^\infty }(0, T;{L^6}(\Omega)), \hfill \\ {\eta ^\varepsilon } \to \eta \quad {\text{ weakly}} {\text{star}} {\text{in}} {H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^2}(0, T;W), \hfill \\ \end{gathered}$

as $\varepsilon \searrow 0$, at least for a subsequence. By accounting for (3.9) and the Lipschitz continuity of g and g′, it is straightforward to see that ($\xi, \eta$) is a solution to problem (3.1)-(3.4). This completes the proof.

We are now prepared to show that S is directionally differentiable. We have the following result:

T_HEOREM 3.2: Suppose that the general hypotheses (A1)-(A5) are satisfied, and let $\bar u \in {U_{{\text{ad}}}}$ be given and $(\bar \rho, \bar \mu)=S(\bar u)$. Moreover, let $h \in X$ be a function such that $\bar u + \bar \lambda h \in {U_{{\text{ad}}}}$ for some $\bar \lambda$ > 0. Then the directional derivative $\delta S(\bar u; h)$ of S at $\bar u$ in the direction h exists in the space ($Y, {\left\| \cdot \right\|_Y}$), and we have $\delta S(\bar u; h)=(\xi, \eta)$, where ($\xi, \eta$) is the unique solution to the linearized system (3.1)-(3.4).

P_ROOF: We have $\bar u + \lambda h \in {U_{{\text{ad}}}}$ for $0 < \lambda \leq \bar \lambda$, since U_ad is convex. We put, for any such $\lambda$,

${u^\lambda }:=\bar u + \lambda h, \qquad ({\rho ^\lambda }, {\mu ^\lambda }):=S({u^\lambda }), \qquad {y^\lambda }:={\rho ^\lambda }-\bar \rho-\lambda \xi, \qquad {z^\lambda }:={\mu ^\lambda }-\bar u-\lambda \eta.$

Notice that (${\rho ^\lambda }, {\mu ^\lambda }$) and ($\bar \rho, \bar \mu$) fulfill the global bounds (2.16), (2.17), and (2.31), and that $({y^\lambda }, {z^\lambda }) \in Y$ 2 Y for all $\lambda \in \left[{0, \bar \lambda } \right]$. Moreover, by virtue of Theorem 2.2, we have the estimate

${\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{H^1}(0, t;H) \cap {L^\infty }(0, t;{L^6}(\Omega))}}{\mkern 1mu} + {\left\| {{\mu ^\lambda }-\bar \mu } \right\|_{{H^1}(0, t;H) \cap {L^\infty }(0, t;V) \cap {L^2}(0, t;W)}}{\mkern 1mu} \leq K_2^*{\mkern 1mu} \lambda {\left\| h \right\|_{{L^2}(0, t;H)}}{\mkern 1mu}.$

(3.30)

We also notice that, owing to (2.16) and the assumptions on F and g, it follows from Taylor’s theorem that

$\left| {F'({\rho ^\lambda })-F'(\bar \rho)-\lambda F''(\bar \rho)\xi } \right| \leq c\left| {{{\text{y}}^\lambda }} \right|{\mkern 1mu} + {\mkern 1mu} c{\left| {{\rho ^\lambda }-\bar \rho } \right|^2}\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} {\text{ }}Q,$

(3.31)

$\left| {g({\rho ^\lambda })-g(\bar \rho)-\lambda g'(\bar \rho)\xi } \right| \leq c\left| {{{\text{y}}^\lambda }} \right|{\mkern 1mu} + {\mkern 1mu} c{\left| {{\rho ^\lambda }-\bar \rho } \right|^2}\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} {\text{ }}Q,$

(3.32)

$\left| {g'({\rho ^\lambda })-g'(\bar \rho)-\lambda g''(\bar \rho)\xi } \right| \leq c\left| {{{\text{y}}^\lambda }} \right|{\mkern 1mu} + {\mkern 1mu} c{\left| {{\rho ^\lambda }-\bar \rho } \right|^2}\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} {\text{ }}Q,$

(3.33)

where, here and in the remainder of the proof, we denote by c constants that may depend on the data of the system but not on $\lambda \in \left[{0, \bar \lambda } \right]$; the actual value of c may change within formulas and lines. Moreover, by the Fréchet differentiability of B (recall assumption (A4)(vi) and the fact that, for $\overline v, v\in L^2(Q)$, the restrictions of B[v] and DB[$\overline v$](v) to Q_t depend only on v|Q_t), we have (cf. (3.30))

${\left\| {B[{\rho ^\lambda }]-B[\bar \rho]-\lambda {\mkern 1mu} DB[\bar \rho](\xi)} \right\|_{{L^2}({Q_t})}} \leq c{\mkern 1mu} {\left\| {{y^\lambda }} \right\|_{{L^2}({Q_t})}} + R\left({\lambda {\mkern 1mu} {{\left\| h \right\|}_{{L^2}({Q_t})}}} \right),$

(3.34)

with a function R : (0; +∞)→(0; +∞) satisfying $\mathop {\lim }\limits_{\sigma \searrow 0} R(\sigma)/\sigma=0$. As we want to prove that $\delta S(\bar u; h)=(\xi, \eta)$, according to the definition of directional differentiability, we need to show that

$\begin{gathered} 0=\mathop {\lim }\limits_{\lambda \searrow 0} {\mkern 1mu} \frac{{{{\left\| {S(\bar u + \lambda h)-S(\bar u)-\lambda {\mkern 1mu} (\xi, \eta)} \right\|}_Y}}}{\lambda } \\=\mathop {\lim }\limits_{\lambda \searrow 0} {\mkern 1mu} \frac{{{{\left\| {{{\text{y}}^\lambda }} \right\|}_{{H^1}(0, T;H)}}{\mkern 1mu} + {{\left\| {{{\text{z}}^\lambda }} \right\|}_{{L^\infty }(0, T;H) \cap {L^2}(0, T;V)}}}}{\lambda }{\mkern 1mu} {\mkern 1mu}. \\ \end{gathered}$

(3.35)

To begin with, using the state system (1.2)-(1.5) and the linearized system (3.1)-(3.4), we easily verify that for $0 < \lambda \leq \bar \lambda$ the pair (${z^\lambda }, {y^\lambda }$) is a strong solution to the system

$\begin{gathered} (1 + 2g(\bar \rho))z_t^\lambda + g'(\bar \rho){{\bar \rho }_t}{{\text{z}}^\lambda }{\mkern 1mu} + \bar \mu {\mkern 1mu} g'(\bar \rho)z_t^\lambda-\Delta {z^\lambda } \hfill \\=-{\mkern 1mu} 2\left({g({\rho ^\lambda })-g(\bar \rho)} \right)\left({\mu _t^\lambda-{{\bar \mu }_t}} \right){\mkern 1mu}-{\mkern 1mu} 2{{\bar \mu }_t}\left({g({\rho ^\lambda })-g(\bar \rho)-\lambda g'(\bar \rho)\xi } \right) \hfill \\ \quad-\bar \mu {{\bar \rho }_t}\left({g'({\rho ^\lambda })-g'(\bar \rho)-\lambda g''(\bar \rho)\xi } \right)-\bar \mu \left({g'({\rho ^\lambda })-g'(\bar \rho)} \right)\left({\rho _t^\lambda-{{\bar \rho }_t}} \right) \hfill \\ \quad {\mkern 1mu}-\left({{\mu ^\lambda }-\bar \mu } \right)\left[{(g'({\rho ^\lambda })-g'(\bar \rho)){\mkern 1mu} {{\bar \rho }_t} + g'({\rho ^\lambda })\left({\rho _t^\lambda-{{\bar \rho }_t}} \right)} \right]\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in }}{\mkern 1mu} Q, \hfill \\ \end{gathered}$

(3.36)

$\begin{gathered} y_t^\lambda=-\left({F'({\rho ^\lambda })-F'(\bar \rho)-\lambda F''(\bar \rho)\xi } \right)-\left({B[{\rho ^\lambda }]-B[\bar \rho]-\lambda DB[\bar \rho](\xi)} \right) \\ + {\mkern 1mu} g'(\bar \rho){{\text{z}}^\lambda } + \bar \mu \left({g'({\rho ^\lambda })-g'(\bar \rho)-\lambda g''(\bar \rho)\xi } \right) \\ + \left({{\mu ^\lambda }-\bar \mu } \right)\left({g'({\rho ^\lambda })-g'(\bar \rho)} \right)\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.in}}{\mkern 1mu} {\text{ }}Q, \\ \end{gathered}$

(3.37)

${\partial _{\text{n}}}{z^\lambda }=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{. on }}{\mkern 1mu} \Sigma,$

(3.38)

${z^\lambda }(0)={y^\lambda }(0)=0\quad {\text{a}}{\text{.}}{\mkern 1mu} {\text{e}}{\text{.}} {\text{in }}{\mkern 1mu} \Omega.$

(3.39)

In the following, we make repeated use of the mean value theorem and of the global estimates (2.16), (2.17), (2.31), and (3.30), without further reference. For the sake of a better readability, we will omit the superscript $\lambda$ of the quantities ${y^\lambda }, {z^\lambda }$ during the estimations, writing it only at the end of the respective estimates.

S_TEP 1: Let $t\in (0, T]$ be fixed. First, observe that

${\partial _t}((\frac{1}{2} + g(\bar \rho)){z^2})=(1 + 2{\mkern 1mu} g(\bar \rho)){\mkern 1mu} z{\mkern 1mu} {z_t} + g'(\bar \rho){{\bar \rho }_t}{\mkern 1mu} {z^2}{\mkern 1mu}.$

Hence, multiplication of (3.36) by z and integration over Q_t yields the estimate

$\int_\Omega {} (\frac{1}{2} + g(\bar \rho (t))){z^2}(t)dx + \int_{{Q_t}} {} |\nabla z{|^2}dx ds \leq c\sum\limits_{j=1}^7 | {I_j}|{\mkern 1mu},$

(3.40)

where the quantities I_j, 1≤j≤7, are specified and estimated as follows: at first, Young’s inequality shows that, for every ${\lambda > 0}$ (to be chosen later),

${I_1}:={\mkern 1mu}-\int_{{Q_t}} {} \bar \mu g'(\bar \rho){\mkern 1mu} {y_t}{\mkern 1mu} z dx ds \leq \gamma \int_{{Q_t}} {} y_t^2dx ds + \frac{c}{\gamma }\int_{{Q_t}} {} {z^2}dx ds{\mkern 1mu}.$

(3.41)

Moreover, we have, by H¨older’s and Young’s inequalities and (3.30),

$\begin{gathered} {I_2}:={\mkern 1mu}-2\int_{{Q_t}} {} \left({g({\rho ^\lambda })-g(\bar \rho)} \right)\left({\mu _t^\lambda-{{\bar \mu }_t}} \right)z dx ds \\ \leq c\int_0^t {} {\left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_6}{\mkern 1mu} {\left\| {\mu _t^\lambda (s)-{{\bar \mu }_t}(s)} \right\|_2}{\mkern 1mu} {\left\| {z(s)} \right\|_3} \\ \leq c{\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{L^\infty }(0, t;{L^6}(\Omega))}}{\left\| {{\mu ^\lambda }-\bar \mu } \right\|_{{H^1}(0, t;H)}}{\mkern 1mu} {\left\| z \right\|_{{L^2}(0, T;V)}} \leq \gamma {\mkern 1mu} \left\| z \right\|_{{L^2}(0, T;V)}^2 + \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.42)

Next, we employ (3.32), the Hälder and Young inequalities, and (3.30), to infer that

$\begin{gathered} {I_3}:=-2\int_{{Q_t}} {} {{\bar \mu }_t}\left({g({\rho ^\lambda })-g(\bar \rho)-\lambda g'(\bar \rho)\xi } \right)zdx ds \\ \leq c\int_{{Q_t}} {} |{{\bar \mu }_t}|\left({|y|{\mkern 1mu} + {\mkern 1mu} |{\rho ^\lambda }-\bar \rho {|^2}} \right){\mkern 1mu} |z|dx ds \\ \leq c\int_0^t {} {\left\| {{{\bar \mu }_t}(s)} \right\|_6}\left({{{\left\| {y(s)} \right\|}_2}{\mkern 1mu} {{\left\| {z(s)} \right\|}_3}{\mkern 1mu} + {\mkern 1mu} \left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_6^2{\mkern 1mu} {{\left\| {z(s)} \right\|}_2}} \right)ds \\ \leq \gamma \int_0^t {} \left\| {z(s)} \right\|_V^2ds{\mkern 1mu} + {\mkern 1mu} \frac{c}{\gamma }\int_0^t {} \left\| {{{\bar \mu }_t}(s)} \right\|_V^2{\mkern 1mu} \left\| {y(s)} \right\|_H^2ds + c\int_0^t {} \left\| {{{\bar \mu }_t}(s)} \right\|_V^2{\mkern 1mu} \left\| {z(s)} \right\|_H^2ds + {\mkern 1mu} c\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{L^\infty }(0, t;V)}^4 \\ \leq \gamma \int_0^t {} \left\| {z(s)} \right\|_V^2ds + {\mkern 1mu} (1 + \frac{c}{\gamma })\int_0^t {} \left\| {{{\bar \mu }_t}(s)} \right\|_V^2(\left\| {y(s)} \right\|_H^2{\mkern 1mu} + \left\| {{\mkern 1mu} z(s)} \right\|_H^2)ds + c{\mkern 1mu} {\lambda ^4}, \\ \end{gathered}$

(3.43)

where we observe that, in view of (2.17), the mapping ${\mkern 1mu} s \mapsto \left\| {{{\bar \mu }_t}(s)} \right\|_V^2{\mkern 1mu}$ belongs to L¹(0, T). Likewise, utilizing (2.17), (3.33), (3.30), and the Hälder and Young inequalities, it is straightforward to deduce that

$\begin{gathered} {I_4}:=-\int_{{Q_t}} {} \bar \mu {{\bar \rho }_t}\left({g'({\rho ^\lambda })-g'(\bar \rho)-\lambda g''(\bar \rho)\xi } \right)zdx ds \leq c\int_{{Q_t}} {} \left({|y| + |{\rho ^\lambda }-\bar \rho {|^2}} \right)|z|dx ds \\ \leq c\int_{{Q_t}} {} \left({{y^2} + {z^2}} \right)dx ds + c\int_0^t {} \left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_4^2{\mkern 1mu} {\left\| {z(s)} \right\|_2}ds{\mkern 1mu} \leq c\int_{{Q_t}} {} \left({{y^2} + {z^2}} \right)dx ds + c{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.44)

In addition, arguing similarly, we have

$\begin{gathered} {I_5}:={\mkern 1mu}-\int_{{Q_t}} {} \bar \mu \left({g'({\rho ^\lambda })-g'(\bar \rho)} \right)\left({{\rho ^\lambda }_t-{{\bar \rho }_t}} \right)zdx ds \\ \leq c\int_0^t {} {\left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_6}{\mkern 1mu} {\left\| {{\rho ^\lambda }_t(s)-{{\bar \rho }_t}(s)} \right\|_2}{\mkern 1mu} {\left\| {z(s)} \right\|_3}ds \\ \leq c{\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{L^\infty }(0, t;{L^6}(\Omega))}}{\mkern 1mu} {\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{H^1}(0, t;H)}}{\mkern 1mu} {\left\| z \right\|_{{L^2}(0, t;V)}} \leq \gamma \int_0^t {} \left\| {z(s)} \right\|_V^2ds{\mkern 1mu} + \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}, \\ \end{gathered}$

(3.45)

as well as

$\begin{gathered} {I_6}:=-\int_{{Q_t}} {} {{\bar \rho }_t}\left({{\mu ^\lambda }-\bar \mu } \right)\left({g'({\rho ^\lambda })-g'(\bar \rho)} \right)zdx ds \leq c\int_{{Q_t}} {} \left| {{\mu ^\lambda }-\bar \mu } \right|\left| {{\rho ^\lambda }-\bar \rho } \right||z|dx ds \\ \leq c{\mkern 1mu} \int_0^t {} {\left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_6}{\left\| {{\mu ^\lambda }(s)-\bar \mu (s)} \right\|_3}{\left\| {z(s)} \right\|_2}ds \leq c\int_{{Q_t}} {} {z^2}dx ds + c{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.46)

Finally, we find that

$\begin{gathered} {I_7}:=-\int_{{Q_t}} {} \left({{\mu ^\lambda }-\bar \mu } \right)g'({\rho ^\lambda })\left({{\rho ^\lambda }_t-{{\bar \rho }_t}} \right)zdx ds \\ \leq c\int_0^t {} {\left\| {{\mu ^\lambda }(s)-\bar \mu (s)} \right\|_6}{\mkern 1mu} {\left\| {\rho _t^\lambda (s)-{{\bar \rho }_t}(s)} \right\|_2}{\left\| {z(s)} \right\|_3}ds \\ \leq c{\left\| {{\mu ^\lambda }-\bar \mu } \right\|_{{L^\infty }(0, t;V)}}{\mkern 1mu} {\left\| {{\rho ^\lambda }-\bar \rho } \right\|_{{H^1}(0, t;H)}}{\left\| z \right\|_{{L^2}(0, t;V)}}{\mkern 1mu} \leq \gamma \int_0^t {} \left\| {z(s)} \right\|_V^2ds + \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.47)

In conclusion, combining the estimates (3.40)-(3.47), and choosing $\gamma=\frac 18$, we have shown that

$\begin{gathered} \frac{1}{2}{\mkern 1mu} \left\| {{{\text{z}}^\lambda }(t)} \right\|_H^2 + \frac{1}{2}\int_0^t {} \left\| {{{\text{z}}^\lambda }(s)} \right\|_V^2ds{\mkern 1mu} \leq {\mkern 1mu} \frac{1}{8}\int_{{Q_t}} {} {\left| {y_t^\lambda } \right|^2}dxds{\mkern 1mu} + {\mkern 1mu} c{\mkern 1mu} {\lambda ^4} \\ + {\mkern 1mu} c\int_0^t {} \left({1 + \left\| {{{\bar \mu }_t}(s)} \right\|_V^2} \right)\left({\left\| {{{\text{y}}^\lambda }(s)} \right\|_H^2{\mkern 1mu} + \left\| {{{\text{z}}^\lambda }(s)} \right\|_H^2} \right)ds{\mkern 1mu}. \\ \end{gathered}$

(3.48)

S_TEP 2: Let $t\in (0, T]$ be fixed. We add y to both sides of (3.37), multiply the resulting identity by y_t, and integrate over Q_t to obtain

$\int_{{Q_t}} {} y_t^2dx ds + {\mkern 1mu} \frac{1}{2}\left\| {y(t)} \right\|_H^2{\mkern 1mu} \leq {\mkern 1mu} \sum\limits_{j=1}^6 | {J_j}|{\mkern 1mu},$

(3.49)

where the terms J_j, 1≤j≤6, are specified and estimated as follows: at first, we have, for every $\gamma>0$ (to be specified later),

${J_1}:=\int_{{Q_t}} {} y{\mkern 1mu} {y_t}dx ds \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} y_t^2dx ds + {\mkern 1mu} \frac{c}{\gamma }\int_{{Q_t}} {} {y^2}dx ds.$

(3.50)

Then, we employ (2.17), (2.31), (3.30), and (3.31), as well as Hölder’s and Young’s inequalities, to obtain the estimate

$\begin{gathered} {J_2}:={\mkern 1mu}-\int_{{Q_t}} {} \left({F'({\rho ^\lambda })-F'(\bar \rho)-\lambda F''(\bar \rho)\xi } \right){y_t}dx ds{\mkern 1mu} \leq {\mkern 1mu} c\int_{{Q_t}} {} \left({|y| + |{\rho ^\lambda }-\bar \rho {|^2}} \right)|{y_t}|dx ds{\mkern 1mu} \\ \leq {\mkern 1mu} c\int_0^t {} \left({{{\left\| {y(s)} \right\|}_2}{\mkern 1mu} + {\mkern 1mu} \left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_4^2} \right){\left\| {{y_t}(s)} \right\|_2}ds{\mkern 1mu} \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} y_t^2dx ds{\mkern 1mu} + {\mkern 1mu} \frac{c}{\gamma }\int_{{Q_t}} {} {y^2}dx ds{\mkern 1mu} + {\mkern 1mu} \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.51)

By the same token, this time invoking (3.33), we find that

${J_3}:=\int_{{Q_t}} {} \bar \mu \left({g'({\rho ^\lambda })-g'(\bar \rho)-\lambda g''(\bar \rho)\xi } \right){y_t}dx ds \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} y_t^2dx ds{\mkern 1mu} + {\mkern 1mu} \frac{c}{\gamma }\int_{{Q_t}} {} {y^2}dx ds + {\mkern 1mu} \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}.$

(3.52)

Moreover, we obviously have

${J_4}:=\int_{{Q_t}} {} g'(\bar \rho){\mkern 1mu} z{\mkern 1mu} {y_t}dx ds{\mkern 1mu} \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} y_t^2dx ds + {\mkern 1mu} \frac{c}{\gamma }\int_{{Q_t}} {} {z^2}dx ds{\mkern 1mu}.$

(3.53)

Also, using (3.30) and the global bounds once more, we obtain that

$\begin{gathered} {J_5}:=\int_{{Q_t}} {} ({\mu ^\lambda }-\bar \mu){\mkern 1mu} (g'({\rho ^\lambda })-g'(\bar \rho)){\mkern 1mu} {y_t}dx ds \\ \leq {\mkern 1mu} c{\mkern 1mu} \int_0^t {} {\left\| {{\mu ^\lambda }(s)-\bar \mu (s)} \right\|_6}{\mkern 1mu} {\left\| {{\rho ^\lambda }(s)-\bar \rho (s)} \right\|_3}{\mkern 1mu} {\left\| {{y_t}(s)} \right\|_2}ds \leq \gamma \int_{{Q_t}} {} y_t^2dxdx + {\mkern 1mu} \frac{c}{\gamma }{\mkern 1mu} {\lambda ^4}{\mkern 1mu}. \\ \end{gathered}$

(3.54)

Finally, invoking (3.34) and Young’s inequality, we have the estimate

$\begin{gathered} {J_6}:={\mkern 1mu}-\int_{{Q_t}} {} \left({B[{\rho ^\lambda }]-B[\bar \rho]-\lambda DB[\bar \rho](\xi)} \right){y_t}dx ds \\ \leq {\left\| {B[{\rho ^\lambda }]-B[\bar \rho]-\lambda DB[\bar \rho](\xi)} \right\|_{{L^2}({Q_t})}}{\left\| {{y_t}} \right\|_{{L^2}({Q_t})}} \\ \leq {\mkern 1mu} \gamma \int_{{Q_t}} {} y_t^2dx ds + \frac{c}{\gamma }\left\| y \right\|_{{L^2}({Q_t})}^2 + \frac{c}{\gamma }{\mkern 1mu} {\left({R\left({\lambda {{\left\| h \right\|}_{{L^2}(Q)}}} \right)} \right)^2}{\mkern 1mu}. \\ \end{gathered}$

(3.55)

Thus, combining the estimates (3.49)-(3.55), and choosing $\gamma=\frac 18$, we have shown that, for every $t\in (0, T]$, we have the estimate

$\frac{1}{4}\int_{{Q_t}} {} |y_t^\lambda {|^2}dx ds + {\mkern 1mu} \frac{1}{2}{\mkern 1mu} \left\| {{y^\lambda }(t)} \right\|_H^2{\mkern 1mu} \leq {\mkern 1mu} c\left({\int_0^t {} \left\| {{y^\lambda }(s)} \right\|_H^2ds + {\lambda ^4}{\mkern 1mu} + {{\left({R\left({\lambda {{\left\| h \right\|}_{{L^2}(Q)}}} \right)} \right)}^2}} \right){\mkern 1mu}.$

(3.56)

S_TEP 3: We now add the estimates (3.48) and (3.56). It follows that, with suitable global constants c₁ > 0 and c₂ > 0, we have for every $t\in (0, T]$ the estimate

$\begin{gathered} \left\| {{{\text{z}}^\lambda }(t)} \right\|_H^2{\mkern 1mu} + \left\| {{{\text{z}}^\lambda }} \right\|_{{L^2}(0, t;V)}^2{\mkern 1mu} + \left\| {{{\text{y}}^\lambda }(t)} \right\|_H^2{\mkern 1mu} + \left\| {y_t^\lambda } \right\|_{{L^2}(0, t;H)}^2 \hfill \\ \leq {\mkern 1mu} {c_1}{\mkern 1mu} Z(\lambda){\mkern 1mu} + {\mkern 1mu} {c_2}\int_0^t {} \left({1 + \left\| {{{\bar \mu }_t}(s)} \right\|_V^2} \right)\left({\left\| {{y^\lambda }(s)} \right\|_H^2{\mkern 1mu} + \left\| {{z^\lambda }(s)} \right\|_H^2} \right)ds, \hfill \\ \end{gathered}$

(3.57)

where we have defined, for $\lambda > 0$, the function Z by

$Z(\lambda):={\mkern 1mu} {\lambda ^4}{\mkern 1mu} + {\mkern 1mu} {(R(\lambda {\mkern 1mu} {\left\| h \right\|_{{L^2}(\Omega)}}))^2}{\mkern 1mu}.$

(3.58)

Recalling that the mapping ${\mkern 1mu} s \mapsto \left\| {{{\bar \mu }_t}(s)} \right\|_V^2{\mkern 1mu}$ belongs to L¹(0, T), we conclude from Gronwall’s lemma that, for every $t\in (0, T]$,

$\left\| {{y^\lambda }} \right\|_{{H^1}(0, t;H)}^2{\mkern 1mu} + {\mkern 1mu} \left\| {{z^\lambda }} \right\|_{{L^\infty }(0, t;H) \cap {L^2}(0, t;V)}^2{\mkern 1mu} \leq {\mkern 1mu} {c_1}{\mkern 1mu} Z(\lambda)\exp \left({{c_2}\int_0^T \left({1 + \left\| {{{\bar \mu }_t}(s)} \right\|_V^2} \right)ds} \right){\mkern 1mu} \leq {\mkern 1mu} c{\mkern 1mu} Z\left(\lambda \right){\mkern 1mu}.$

(3.59)

Since $\lim_{\lambda\searrow 0}\, Z(\lambda)/\lambda^2=0\,$ (recall (3.34)), we have finally shown the validity of (3.35). This concludes the proof of the assertion.

We are now in the position to derive the following result.

C_OROLLARY 3.3: Let the general hypotheses (A1)-(A5) be fulfilled and assume that $\bar u \in {U_{{\text{ad}}}}$ is a solution to the control problem (CP) with associated state $(\bar \rho, \bar \mu)=S(\bar u)$. Then we have, for every $v \in {U_{{\text{ad}}}}$,

${\beta _1}\int_Q {} (\bar \rho-{\rho _Q}){\mkern 1mu} \xi dx dt + {\mkern 1mu} {\beta _2}\int_Q {} (\bar \mu-{\mu _Q}){\mkern 1mu} \eta dx dt{\mkern 1mu} + {\mkern 1mu} {\beta _3}\int_Q {} \bar u{\mkern 1mu} (v-\bar u)dx dt \geq {\mkern 1mu} 0{\mkern 1mu},$

(3.60)

where $(\xi, \eta)$ denotes the (unique) solution to the linearized system (3.1)-(3.4) for $h=v-\bar u$.

P_ROOF: Let $v \in {U_{{\text{ad}}}}$ be arbitrary. Then $h=v-\bar u$ is an admissible direction, since $\bar u + \lambda h \in {U_{{\text{ad}}}}$ for $0 < \lambda\le 1$. For any such $\lambda$, we have

$\begin{gathered} 0 \leq \frac{{J(\bar u + \lambda h, S(\bar u + \lambda h))-J(\bar u, S(\bar u))}}{\lambda } \\ \leq {\mkern 1mu} \frac{{J(\bar u + \lambda h, S(\bar u + \lambda h))-J(\bar u, S(\bar u + \lambda h))}}{\lambda }{\mkern 1mu} + {\mkern 1mu} \frac{{J(\bar u, S(\bar u + \lambda h)-J(\bar u, S(\bar u))}}{\lambda }{\mkern 1mu} {\mkern 1mu}. \\ \end{gathered}$

It follows immediately from the definition of the cost functional J that the first summand on the right-hand side of this inequality converges to ${\mkern 1mu} \int_Q {} {\beta _3}{\mkern 1mu} \bar u{\mkern 1mu} hdx dt{\mkern 1mu}$ as $\lambda\searrow 0$. For the second summand, we obtain from Theorem 3.2 that

$\mathop {\lim }\limits_{\lambda \searrow 0} {\mkern 1mu} \frac{{J(\bar u, S(\bar u + \lambda h)-J(\bar u, S(\bar u))}}{\lambda }{\mkern 1mu}={\mkern 1mu} {\beta _1}\int_Q {} (\bar \rho-{\rho _Q})\xi dx dt + {\mkern 1mu} {\beta _2}\int_Q {} (\bar \mu-{\mu _Q})\eta dx dt{\mkern 1mu},$

whence the assertion follows.

---

4. Existence and first-order necessary conditions of optimality

In this section, we derive the first-order necessary conditions of optimality for problem (CP). We begin with an existence result.

T_HEOREM 4.1: Suppose that the conditions (A1)-(A5) are satisfied. Then the problem (CP) has a solution $\bar u \in {U_{{\text{ad}}}}$.

P_ROOF: Let ${\{ {u_n}\} _{n \in \mathbb{N}}} \subset {U_{{\text{ad}}}}$ be a minimizing sequence for (CP), and let ${\{ ({\rho _n}, {\mu _n})\} _{n \in \mathbb{N}}}$ be the sequence of the associated solutions to (1.2)-(1.5). We then can infer from the global estimate (2.17) the existence of a triple ($\bar u, \bar \rho, \bar \mu$) such that, for a suitable subsequence again indexed by n,

$\begin{gathered} {u_n} \to \bar u{\mkern 1mu} \quad {\text{weakly star in }}{\mkern 1mu} {H^1}(0, T;H) \cap {L^\infty }(Q), \hfill \\ {\rho _n} \to \bar \rho {\mkern 1mu} \quad {\text{weakly star in }}{\mkern 1mu} {H^2}(0, T;H) \cap n{W^{1, \infty }}(0, T;{L^\infty }(\Omega)) \cap {H^1}(0, T;V), \hfill \\ {\mu _n} \to \bar \mu {\mkern 1mu} \quad {\text{weakly star in }}{\mkern 1mu} {W^{1, \infty }}(0, T;H) \cap {H^1}(0, T;V) \cap {L^\infty }(0, T;W). \hfill \\ \end{gathered}$

Clearly, we have that $\bar u \in {U_{{\text{ad}}}}$ and, by virtue of the Aubin-Lions lemma (cf. [24, Thm. 5.1, p. 58]) and similar compactness results (cf. [27, Sect. 8, Cor. 4]),

${\rho _n} \to {\text{ }}\bar \rho {\mkern 1mu} \quad {\text{strongly in }}{\mkern 1mu} {L^2}(Q),$

(4.1)

whence also ${\rho _*} \leq {\text{ }}\bar \rho \leq {\rho ^*}$ a. e. in Q and

$\begin{gathered} B[{\rho _n}] \to B[\bar \rho]\quad {\text{strongly in }}{\mkern 1mu} {L^2}(\Omega), \hfill \\ \Phi ({\rho _n}) \to \Phi (\bar \rho)\quad {\text{strongly in }}{\mkern 1mu} {L^2}(Q), \quad {\text{for }}{\mkern 1mu} \Phi \in \{ F', g, g'\}, \hfill \\ \end{gathered}$

thanks to the general assumptions on B, F and g, as well as the strong convergence

${\mu _n} \to {\text{ }}\bar \mu {\mkern 1mu} \quad {\text{strongly in }}{\mkern 1mu} {C^0}([0, T];{C^0}(\bar \Omega))={C^0}(\bar Q).$

(4.2)

From this, we easily deduce that

$\begin{gathered} g({\rho _n}){\mkern 1mu} {\partial _t}{\mu _n} \to g(\bar \rho){\mkern 1mu} {\partial _t}\bar \mu \quad {\text{weakly in }}{\mkern 1mu} {L^1}(Q), \hfill \\ {\mu _n}{\mkern 1mu} g'({\rho _n}){\mkern 1mu} {\partial _t}{\rho _n} \to \bar \mu {\mkern 1mu} g'(\bar \rho){\partial _t}\bar \rho \quad {\text{weakly in }}{\mkern 1mu} {L^1}(Q){\mkern 1mu}. \hfill \\ \end{gathered}$

In summary, if we pass to the limit as n→∞ in the state equations (1.2)-(1.3), written for the triple $(u_n, \rho_n, \mu_n)$, we find that ($\bar \rho, \bar \mu$) satisfies (1.2) and (1.3). Moreover, $\bar \mu \in {L^\infty }(0, T;W)$ satisfies the boundary condition (1.4), and it is easily seen that also the initial conditions (1.5) hold true. In other words, we have $(\bar \rho, \bar \mu)=S(\bar \mu)$, that is, the triple ($\bar u, \bar \rho, \bar \mu$) is admissible for the control problem (CP). From the weak sequential lower semicontinuity of the cost functional J it finally follows that $\bar u$, together with $(\bar \rho, \bar \mu)=S(\bar \mu)$, is a solution to (CP). This concludes the proof.

We now turn our interest to the derivation of first-order necessary optimality conditions for problem (CP). To this end, we generally assume in the following that the hypotheses (A1)-(A5) are satisfied and that $\bar u \in {U_{{\text{ad}}}}$ is an optimal control with associated state ($\bar \rho, \bar \mu$), which has the properties (2.16)-(2.31). We now aim to eliminate $\xi$ and $\eta$ from the variational inequality (3.60). To this end, we employ the adjoint state system associated with (1.2)-(1.5) for $\bar u$, which is formally given by:

$-(1 + 2g(\bar \rho)){\mkern 1mu} {p_t}-g'(\bar \rho){{\bar \rho }_t}{\mkern 1mu} p-\Delta p-g'(\bar \rho){\mkern 1mu} q={\text{ }}{\beta _2}(\bar \mu-{\mu _Q})\quad {\text{in }}{\mkern 1mu} Q,$

(4.3)

$-{q_t} + F''(\bar \rho){\mkern 1mu} q-\bar \mu g''(\bar \rho){\mkern 1mu} q + g'(\bar \rho)\left({{{\bar \mu }_t}{\mkern 1mu} p-\bar \mu {\mkern 1mu} {p_t}} \right) + DB{[\bar \rho]^*}(q)={\text{ }}{\beta _1}(\bar \rho-{\rho _Q})\quad {\text{in }}{\mkern 1mu} Q,$

(4.4)

${\partial _{\text{n}}}p=0\quad {\text{on }}{\mkern 1mu} \Sigma,$

(4.5)

$p(T)=q(T)=0\quad {\text{in }}{\mkern 1mu} \Omega {\mkern 1mu}.$

(4.6)

In (4.4), $DB{[\bar \rho]^*} \in L({L^2}(Q), {L^2}(Q))$ denotes the adjoint operator associated with the operator $DB{[\bar \rho]} \in L({L^2}(Q), {L^2}(Q))$, thus defined by the identity

$\int_Q {} DB{[\bar \rho]^*}(v){\mkern 1mu} wdx dt=\int_Q {} v{\mkern 1mu} DB[\bar \rho](w)dx dt\quad \forall {\mkern 1mu} v, w \in {L^2}(Q){\mkern 1mu}.$

(4.7)

As, for every $v\in L^2(Q)$, the restriction of $DB[\bar \rho](v)$ to Q_t depends only on vjQt, it follows that, for every $w\in L^2(Q)$, the restriction of $DB{[\bar \rho]^*}(w)$ to Q^t=Ω×(t, T) (see (1.14)) depends only on w|Q_t. Moreover, (2.8) implies that

${\left\| {DB{{[\bar \rho]}^*}(w)} \right\|_{{L^2}({Q^t})}} \leq {C_B}{\left\| w \right\|_{{L^2}({Q^t})}}\quad \forall {\mkern 1mu} w \in {L^2}(Q).$

(4.8)

We also note that in the case of the integral operator (1.9) it follows from Fubini’s theorem that $DB{[\bar \rho]^*}=DB[\bar \rho]=B$.

We have the following existence and uniqueness result for the adjoint system.

T_HEOREM 4.2: Suppose that (A1)-(A5) are fulfilled, and assume that $\bar u \in {U_{{\text{ad}}}}$ is a solution to the control problem (CP) with associated state $(\bar \rho, \bar \mu)=S(\bar u)$. Then the adjoint system (4.3)-(4.6) has a unique solution (p, q) satisfying

$p \in {H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^2}(0, T;W) \qquad and \qquad q \in {H^1}(0, T;H){\mkern 1mu}.$

(4.9)

P_ROOF: Besides of Young’s inequality, we make repeated use of the global estimates (2.16)-(2.17) and (2.31) for ${\bar \rho }$ and ${\bar \mu }$, without further reference. Moreover, we denote by c different positive constants that may depend on the given data of the state system and of the control problem; the meaning of c may change between and even within lines.

We first prove uniqueness. Thus, we replace the right-hand sides of (4.3) and (4.4) by 0 and prove that (p, q)=(0, 0). We add p to both sides of (4.3) and multiply by-p_t. At the same time, we multiply (4.4) by q. Then we add the resulting equalities and integrate over Q^t=Ω×(t, T). As g is nonnegative, and thanks to (2.8), we obtain that

$\begin{gathered} \int_{{Q^t}} {} |{p_t}{|^2}dx ds{\mkern 1mu} + {\mkern 1mu} \frac{1}{2}{\mkern 1mu} \left\| {p(t)} \right\|_V^2{\mkern 1mu} + {\mkern 1mu} \frac{1}{2}\int_\Omega {} |q(t){|^2}dx \\ \leq \int_{{Q^t}} {} (-p-g'(\bar \rho){{\bar \rho }_t}{\mkern 1mu} p-g'(\bar \rho){\mkern 1mu} q){\mkern 1mu} {p_t}dx ds \\ + \int_{{Q^t}} {} ((\bar \mu {\mkern 1mu} g''(\bar \rho)-F''(\bar \rho)){\mkern 1mu} q + \bar \mu g'(\bar \rho){\mkern 1mu} {p_t}-DB[\bar \rho](q)){\mkern 1mu} qdx ds-\int_{{Q^t}} {} g'(\bar \rho){\mkern 1mu} {{\bar \mu }_t}{\mkern 1mu} p{\mkern 1mu} qdx ds \\ \leq \frac{1}{2}\int_{{Q^t}} {} |{p_t}{|^2}dx ds + c\int_{{Q^t}} {} ({p^2} + {q^2})dx ds + c\int_{{Q^t}} {} |{{\bar \mu }_t}|{\mkern 1mu} |p|{\mkern 1mu} |q|dx ds{\mkern 1mu}. \\ \end{gathered}$

The last integral is estimated as follows: employing the Hölder, Sobolev and Young inequalities, we have

$\int_{{Q^t}} {} |{{\bar \mu }_t}|{\mkern 1mu} |p|{\mkern 1mu} |q|dx ds \leq \int_t^T {} {\left\| {{{\bar \mu }_t}(s)} \right\|_3}{\left\| {p(s)} \right\|_6}{\left\| {q(s)} \right\|_2}ds \leq c\int_t^T {} (\left\| {{{\bar \mu }_t}(s)} \right\|_V^2\left\| {p(s)} \right\|_V^2 + \left\| {q(s)} \right\|_H^2)ds{\mkern 1mu}.$

As the function $s \mapsto \left\| {{{\bar \mu }_t}(s)} \right\|_V^2$ belongs to L¹(0, T), we can apply the backward version of Gronwall’s lemma to conclude that (p, q)=(0, 0).

The existence of a solution to (4.3)-(4.6) is proved in several steps.

S_TEP 1: We approximate ${\bar \rho }$ and ${\bar \mu }$ by functions ${\rho ^\varepsilon }, {\mu ^\varepsilon } \in {C^\infty }(\bar Q)$ satisfying (3.8)-(3.9) and look for a solution (${p^\varepsilon }, {q^\varepsilon }$) to the following problem:

$-(1 + 2g({\rho ^\varepsilon }))p_t^\varepsilon-g'(\bar \rho){{\bar \rho }_t}{p^\varepsilon }-\Delta {p^\varepsilon }-g'(\bar \rho){q^\varepsilon }={\beta _2}(\bar \mu-{\mu _Q})\quad {\text{in }}{\mkern 1mu} Q,$

(4.10)

$\begin{gathered}-q_t^\varepsilon-\varepsilon {\mkern 1mu} \Delta {q^\varepsilon } + F''(\bar \rho){\mkern 1mu} {q^\varepsilon }-\bar \mu {\mkern 1mu} g''(\bar \rho){\mkern 1mu} {q^\varepsilon } + g'({\rho ^\varepsilon })\left({\mu _t^\varepsilon {p^\varepsilon }-{\mu ^\varepsilon }p_t^\varepsilon } \right) + DB{[\bar \rho]^*}({q^\varepsilon }) \\={\beta _1}({q^\varepsilon }-{\rho _Q})\quad {\text{in }}{\mkern 1mu} Q, \\ \end{gathered}$

(4.11)

${\partial _{\text{n}}}{p^\varepsilon }={\partial _{\text{n}}}{q^\varepsilon }=0\quad {\text{on }}{\mkern 1mu} \Sigma,$

(4.12)

${p^\varepsilon }(T)={q^\varepsilon }(T)=0\quad {\text{in }}{\mkern 1mu} \Omega {\mkern 1mu}.$

(4.13)

We prove that this problem has a unique solution satisfying

${p^\varepsilon }, {q^\varepsilon } \in {H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^2}(0, T;W){\mkern 1mu}.$

(4.14)

To this end, we present (4.10)-(4.12) as an abstract backward equation, namely,

$-\frac{d}{{dt}}{\mkern 1mu} ({p^\varepsilon }, {q^\varepsilon })(t) + {A^\varepsilon }(t){\mkern 1mu} ({p^\varepsilon }, {q^\varepsilon })(t) + ({R^\varepsilon }({p^\varepsilon }, {q^\varepsilon }))(t)={f^\varepsilon }(t),$

(4.15)

in the framework of the Hilbert triplet (V, H, V′), where

$V:=V \times V{\text{ and }}H:=H \times H{\mkern 1mu}.$

Notice that (4.15), together with the regularity (4.14), means that

$\begin{gathered} \hfill-{((p_t^\varepsilon, q_t^\varepsilon)(t), (v, w))_H} + {a^\varepsilon }(t; ({p^\varepsilon }, {q^\varepsilon })(t), (v, w)) + {(({R^\varepsilon }({p^\varepsilon }, {q^\varepsilon }))(t), (v, w))_H}{\mkern 1mu}={\mkern 1mu} {({f^\varepsilon }(t), (v, w))_H} \\ \hfill {\text{for every }}(v, w) \in V{\text{ and a}}{\text{.}}{\mkern 1mu} {\text{ a}}{\text{.}}\; {\text{ }}t \in (0, T), \\ \end{gathered}$

(4.16)

where ${a^\varepsilon }(t; \cdot, \cdot)$ is the bilinear form associated with the operator ${A^\varepsilon }(t):V \to V'$; moreover, ${(\cdot, \cdot)_H}$ denotes the inner product in H (equivalent to the usual one) that one has chosen, the embedding $H \subset V'$ being dependent on such a choice. In fact, we will not use the standard inner product of H, which will lead to a nonstandard embedding $H \subset V'$. We aim at applying first [1, Thm. 3.2, p. 256], in order to find a unique weak solution, as we did for the linearized problem; then, we derive the full regularity required in (4.14). We set, for convenience,

${\varphi _\varepsilon }:=\frac{1}{{1 + 2g({\rho ^\varepsilon })}}{\text{ and }}{\psi _\varepsilon }:=\frac{{{\mu ^\varepsilon }g'({\rho ^\varepsilon })}}{{1 + 2g({\rho ^\varepsilon })}}={\varphi _\varepsilon }:{\mkern 1mu} {\mu ^\varepsilon }g'({\rho ^\varepsilon }),$

and choose a constant ${M_\varepsilon }$ such that

${\varphi _\varepsilon } \leq {M_\varepsilon }, \quad |{\psi _\varepsilon }| \leq {M_\varepsilon }, \quad |\nabla {\varphi _\varepsilon }| \leq {M_\varepsilon }, {\text{ and }}|\nabla {\varphi _\varepsilon }| \leq {M_\varepsilon }, \quad {\text{ a}}{\text{. }}{\mkern 1mu} {\text{e}}{\text{.}}\; {\text{ in }}Q.$

Moreover, we introduce three parameters ${\lambda ^\varepsilon }, \lambda _1^\varepsilon, \lambda _2^\varepsilon$, whose values will be specified later on. In order to transform our problem, we compute $p_t^\varepsilon$ from (4.10) and substitute in (4.11). Moreover, we multiply (4.10) by ${\varphi _\varepsilon }$. Finally, we add and subtract the same terms for convenience. Then (4.10)-(4.11) is equivalent to the system

$\begin{gathered}-p_t^\varepsilon-{\varphi _\varepsilon }\Delta {p^\varepsilon } + \lambda _1^\varepsilon {p^\varepsilon }-\lambda _1^\varepsilon {p^\varepsilon }-{\varphi _\varepsilon }g'(\bar \rho){\mkern 1mu} {{\bar \rho }_t}{p^\varepsilon }-{\varphi _\varepsilon }{\mkern 1mu} g'(\bar \rho){q^\varepsilon }{\mkern 1mu}={\varphi _\varepsilon }{\beta _2}(\bar \mu-{\mu _Q}), \hfill \\-q_t^\varepsilon-\varepsilon {\mkern 1mu} \Delta {q^\varepsilon } + {\varphi _\varepsilon }{\mkern 1mu} \Delta {p^\varepsilon } + \lambda _2^\varepsilon {q^\varepsilon }-\lambda _2^\varepsilon {q^\varepsilon } + F''(\bar \rho){\mkern 1mu} {q^\varepsilon }-\bar \mu {\mkern 1mu} g''(\bar \rho){q^\varepsilon } + g'({\rho ^\varepsilon })\mu _t^\varepsilon {p^\varepsilon } \hfill \\ + {\psi _\varepsilon }{\mkern 1mu} (g'(\bar \rho){{\bar \rho }_t}{p^\varepsilon } + g'(\bar \rho){q^\varepsilon } + {\beta _2}(\bar \mu-{\mu _Q})) + DB{[\bar \rho]^*}({q^\varepsilon })={\beta _1}(\bar \rho-{\rho _Q}){\mkern 1mu}. \hfill \\ \end{gathered}$

By observing that

$-{\varphi _\varepsilon }{\mkern 1mu} \Delta {p^\varepsilon }=-{\text{div}}({\varphi _\varepsilon }\nabla {p^\varepsilon }) + \nabla {\varphi _\varepsilon } \cdot \nabla {p^\varepsilon }{\mkern 1mu},$

and that the same holds true with ${\psi _\varepsilon }$ in place of ${\varphi _\varepsilon }$, we see that the latter system, complemented with the boundary condition (4.12), is equivalent to

$\begin{gathered}-\int_\Omega {} q_t^\varepsilon (t){\mkern 1mu} vdx + a_t^\varepsilon (t; {p^\varepsilon }(t), v) + \int_\Omega {} (R_1^\varepsilon ({p^\varepsilon }, {q^\varepsilon }))(t){\mkern 1mu} vdx=\int_\Omega {} {\varphi _\varepsilon }(t){\mkern 1mu} {\beta _2}(\bar \mu-{\mu _Q})(t){\mkern 1mu} vdx \hfill \\-\int_\Omega {} q_t^\varepsilon (t){\mkern 1mu} wdx + a_t^\varepsilon (t; ({p^\varepsilon }(t), {q^\varepsilon }(t)), w) + \int_\Omega {} (R_1^\varepsilon ({p^\varepsilon }, {q^\varepsilon }))(t){\mkern 1mu} wdx \hfill \\=-\int_\Omega {} {\psi _\varepsilon }(t){\mkern 1mu} {\beta _2}(\bar \mu-{\mu _Q})(t){\mkern 1mu} wdx + \int_\Omega {} {\beta _1}(\bar \rho-{\rho _Q})(t){\mkern 1mu} wdx \hfill \\ \end{gathered}$

for every (v, w)∈V and a. a. $t\in(0, T)$, where the forms $a_i^\varepsilon$ are defined below and the operators $R_i^\varepsilon$ account for all the other terms on the left-hand sides of the equations. We set, for every $t\in[0, T]$ and $\hat v, \hat w, v, w\in V$,

$\begin{gathered} a_1^\varepsilon (t; \hat v, v):=\int_\Omega {} ({\varphi _\varepsilon }(t){\mkern 1mu} \nabla \hat v \cdot \nabla v + (\nabla {\varphi _\varepsilon }(t) \cdot \nabla \hat v){\mkern 1mu} v + \lambda _1^\varepsilon \hat v{\mkern 1mu} v)dx, \hfill \\ a_2^\varepsilon (t; (\hat v, \hat w), w):=\int_\Omega {} (\varepsilon {\mkern 1mu} \nabla \hat w \cdot \nabla w-{\psi _\varepsilon }(t){\mkern 1mu} \nabla \hat v \cdot \nabla w-(\nabla {\psi _\varepsilon }(t) \cdot \nabla \hat v){\mkern 1mu} w + \lambda _2^\varepsilon \hat w{\mkern 1mu} w)dx{\mkern 1mu}. \hfill \\ \end{gathered}$

Now, we choose the values of $\lambda _i^\varepsilon$ and of the further parameter $\lambda ^\varepsilon$ in such a way as to guarantee some coerciveness. Putting α :=1/(1 + 2 sup g), we have that

$a_1^\varepsilon (t; v, v) \geq \int_\Omega {} (\alpha {\mkern 1mu} |\nabla v{|^2}-{M_\varepsilon }{\mkern 1mu} |\nabla v|{\mkern 1mu} |v| + \lambda _1^\varepsilon {v^2})dx \geq \int_\Omega {} (\alpha {\mkern 1mu} |\nabla v{|^2}-\frac{\alpha }{2}{\mkern 1mu} |\nabla v{|^2}-\frac{{{M_\varepsilon }}}{{2\alpha }}{v^2} + \lambda _1^\varepsilon {v^2})dx{\mkern 1mu}.$

Therefore, the choice $\lambda _1^\varepsilon :=\frac{\alpha }{2} + \frac{{M_\varepsilon ^2}}{{2\alpha }}$ yields

$a_1^\varepsilon (t; v, v){\mkern 1mu} \geq {\mkern 1mu} \frac{\alpha }{2}{\mkern 1mu} \left\| v \right\|{v^2}\quad {\text{ for every }}v \in V{\text{ and }}t \in [0, T].$

Next, we deal with $a_2^\varepsilon$. We have, for every $v, w\in V$ and $t\in[0, T]$,

$\begin{gathered} a_2^\varepsilon (t; (v, w), w) \geq \int_\Omega {} (\varepsilon |\nabla w{|^2}-{M_\varepsilon }|\nabla v|{\mkern 1mu} |\nabla w|-{M_\varepsilon }|\nabla v|{\mkern 1mu} |w| + \lambda _2^\varepsilon {w^2})dx \hfill \\ \geq \int_\Omega {} (\varepsilon |\nabla w{|^2}-\frac{\varepsilon }{2}|\nabla w{|^2}-\frac{{M_\varepsilon ^2}}{{2\varepsilon }}|\nabla v{|^2}-\frac{{M_\varepsilon ^2}}{{2\varepsilon }}|\nabla v{|^2}-\frac{\varepsilon }{2}|w{|^2} + \lambda _2^\varepsilon {w^2})dx \hfill \\=\int_\Omega {} (\frac{\varepsilon }{2}|\nabla w{|^2} + (\lambda _2^\varepsilon-\frac{\varepsilon }{2}){w^2}-\frac{{M_\varepsilon ^2}}{\varepsilon }{\mkern 1mu} |\nabla v{|^2})dx{\mkern 1mu}, \hfill \\ \end{gathered}$

and the choice $\lambda _2^\varepsilon :=\varepsilon$ leads to

$a_2^\varepsilon (t; (v, w), w){\mkern 1mu} \geq \frac{\varepsilon }{2}\left\| w \right\|_V^2-\frac{{M_\varepsilon ^2}}{\varepsilon }\left\| v \right\|_V^2{\mkern 1mu}.$

Therefore, if we choose $\lambda ^\varepsilon$ such that ${\lambda ^\varepsilon } \, \frac{\alpha }{2}-\frac{{M_\varepsilon ^2}}{\varepsilon } \geq \frac{\varepsilon }{2}$, then we obtain

${\lambda ^\varepsilon }a_1^\varepsilon (t; v, v) + a_2^\varepsilon (t; (v, w), w) \geq \frac{\varepsilon }{2}(\left\| v \right\|_V^2 + \left\| w \right\|_V^2)$

for every (v, w)∈V and $t\in[0, T]$. Hence, if we define ${a^\varepsilon }:[0, T] \times V \times V \to \mathbb{R}$ by setting

${a^\varepsilon }(t; (\hat v, \hat w), (v, w)):=\lambda _1^\varepsilon a_1^\varepsilon (\hat v, v) + a_2^\varepsilon (t; (\hat v, \hat w), w){\mkern 1mu},$

then we obtain a time-dependent continuous bilinear form that is coercive on V (endowed whith its standard norm), uniformly with respect to $t\in[0, T]$. Moreover, $a^\varepsilon$ depends smoothly on t, and (4.10)-(4.12) is equivalent to

$\begin{gathered}-\int_\Omega {} ({\lambda ^\varepsilon }p_t^\varepsilon (t){\mkern 1mu} v + q_t^\varepsilon (t){\mkern 1mu} w)dx + {a^\varepsilon }(t; ({p^\varepsilon }(t), {q^\varepsilon }(t)), (v, w)) \hfill \\ + \int_\Omega {} \{ {\lambda ^\varepsilon }{\mkern 1mu} (R_1^\varepsilon ({p^\varepsilon }, {q^\varepsilon }))(t){\mkern 1mu} v + (R_2^\varepsilon ({p^\varepsilon }, {q^\varepsilon }))(t){\mkern 1mu} w\} dx \hfill \\=\int_\Omega {} (({\lambda ^\varepsilon }{\varphi _\varepsilon }-{\psi _\varepsilon })(t){\beta _2}(\bar \mu-{\mu _Q})(t){\mkern 1mu} v + {\beta _1}{\mkern 1mu} (\bar \rho-{\rho _Q})(t){\mkern 1mu} w \hfill \\ \end{gathered}$

for every (v, w)∈V and a. a. $t\in(0, T)$. Therefore, the desired form (4.16) is achieved if we choose the scalar product in H as follows:

${((\hat v, \hat w), (v, w))_H}:=\int_\Omega {} ({\lambda ^\varepsilon }{\mkern 1mu} \hat v{\mkern 1mu} v + \hat w{\mkern 1mu} w)dx\quad {\text{ for every }}(\hat v, \hat w), (v, w) \in H.$

Notice that this leads to the following nonstandard embedding $H \subset V'$ :

$_{V'}{\left\langle {(\hat v, \hat w), (v, w)} \right\rangle _V}={((\hat v, \hat w), (v, w))_H}={\lambda ^\varepsilon }_{V'}{\left\langle {\hat v, v} \right\rangle _V}{ + _{V'}}{\left\langle {\hat w, w} \right\rangle _V}$

for every $(\hat v, \hat w) \in H$ and $(v, w) \in V$, provided that the embedding $H \subset V'$ is the usual one, i. e., corresponds to the standard inner product of H. As the remainder, given by the terms $R_1^\varepsilon$ and $R_2^\varepsilon$, satisfies the backward analogue of (3.14)-(3.15) (see also (4.8)), the quoted result of ^[1] can be applied, and problem (4.10)-(4.13) has a unique solution satisfying

$({p^\varepsilon }, {q^\varepsilon }) \in {H^1}(0, T;V') \cap {L^2}(0, T;V).$

Moreover, if we move the remainder of (4.15) to the right-hand side, we see that

$-\frac{d}{{dt}}({p^\varepsilon }, {q^\varepsilon }) + {A^\varepsilon }({p^\varepsilon }, {q^\varepsilon }) \in {L^2}(0, T;H).$

Therefore, by also accounting for (4.13), we deduce that $({p^\varepsilon }, {q^\varepsilon }) \in {H^1}(0, T;H)$ as well as ${A^\varepsilon }({p^\varepsilon }, {q^\varepsilon }) \in {L^2}(0, T;H)$. Hence, we have that ${p^\varepsilon }, {q^\varepsilon } \in {L^2}(0, T;W)$, by standard elliptic regularity.

S_TEP 2: We add $p^\varepsilon$ to both sides of (4.10) and multiply by $-p_t^\varepsilon$. At the same time, we multiply (4.11) by $q^\varepsilon$. Then, we sum up and integrate over Q_t. As g≥0, we easily obtain that

$\begin{gathered} \frac{1}{2}\left\| {{p^\varepsilon }(t)} \right\|_V^2 + \int_{{Q^t}} {} |p_t^\varepsilon {|^2}dx ds + \frac{1}{2}\int_\Omega {} |{q^\varepsilon }(t){|^2}dx + \varepsilon \int_{{Q^t}} {} |\nabla {q^\varepsilon }{|^2}dx ds \hfill \\ \leq c\int_{{Q^t}} {} |{p^\varepsilon }|\; |p_t^\varepsilon |dx ds + c\int_{{Q^t}} {} |{q^\varepsilon }|\; |p_t^\varepsilon |dx ds + c\int_{{Q^t}} {} |{q^\varepsilon }{|^2}dx ds \hfill \\ + c\int_{{Q^t}} {} |\mu _t^\varepsilon |{\mkern 1mu} |{p^\varepsilon }|{\mkern 1mu} |{q^\varepsilon }|dx ds + \int_{{Q^t}} {} |DB{[\bar \rho]^*}({q^\varepsilon })|{\mkern 1mu} |{q^\varepsilon }|dx ds + c\left\| {{p^\varepsilon }} \right\|_{{L^2}({Q^t})}^2 + c{\mkern 1mu}. \hfill \\ \end{gathered}$

Just two of the terms on the right-hand side need some treatment. We have

$\begin{gathered} \int_{{Q^t}} {} |\mu _t^\varepsilon |{\mkern 1mu} |{p^\varepsilon }|{\mkern 1mu} |{q^\varepsilon }|dx ds \leq \int_t^T {} {\left\| {\mu _t^\varepsilon (s)} \right\|_3}{\left\| {{p^\varepsilon }(s)} \right\|_6}{\left\| {{q^\varepsilon }(s)} \right\|_2}ds \hfill \\ \leq c\int_t^T {} \left\| {{p^\varepsilon }(s)} \right\|_V^2ds + c\int_t^T {} \left\| {\mu _t^\varepsilon (s)} \right\|_3^2\left\| {{q^\varepsilon }(s)} \right\|_2^2ds{\mkern 1mu}, \hfill \\ \end{gathered}$

and we observe that the function $s \mapsto \left\| {\mu _t^\varepsilon (s)} \right\|_3^2$ belongs to L¹(0, T), by (3.8). Moreover, the Schwarz inequality and (4.8) immediately yield that

$\int_{{Q^t}} {} |DB{[\bar \rho]^*}({q^\varepsilon })|{\mkern 1mu} |{q^\varepsilon }|dx ds \leq {C_B}\left\| {{q^\varepsilon }} \right\|_{{L^2}({Q^t})}^2{\mkern 1mu}.$

Therefore, we can apply the backward version of Gronwall’s lemma to obtain that

${\left\| {{p^\varepsilon }} \right\|_{{H^1}(0, T;H) \cap {L^\infty }(0, T;V)}} + {\left\| {{q^\varepsilon }} \right\|_{{H^1}(0, T;H)}} + {\varepsilon ^{1/2}}{\left\| {{q^\varepsilon }} \right\|_{{L^2}(0, T;V)}} \leq c{\mkern 1mu}.$

(4.17)

By comparison in (4.10), we see that $\Delta {p^\varepsilon }$ is bounded in L²(Q). Hence,

${\left\| {{p^\varepsilon }} \right\|_{{L^2}(0, T;W)}} \leq c{\mkern 1mu}.$

(4.18)

S_TEP 3: We multiply (4.11) by ${-q_t^\varepsilon }$ and integrate over Q_t. We obtain

$\begin{gathered} \int_{{Q^t}} {} |q_t^\varepsilon {|^2}dx ds + \frac{\varepsilon }{2}\int_\Omega | \nabla {q^\varepsilon }(t){|^2}dx \hfill \\ \leq c\int_\Omega {} |{q^\varepsilon }|{\mkern 1mu} |q_t^\varepsilon |dx ds + c\int_t^T {} {\left\| {\mu _t^\varepsilon (s)} \right\|_3}{\mkern 1mu} {\left\| {{p^\varepsilon }(s)} \right\|_6}{\left\| {q_t^\varepsilon (s)} \right\|_2}ds \hfill \\ + c\int_{{Q^t}} {} |p_t^\varepsilon |{\mkern 1mu} |q_t^\varepsilon |dx ds + \int_{{Q^t}} {} |DB{[\bar \rho]^*}({q^\varepsilon })|{\mkern 1mu} |q_t^\varepsilon |dx ds{\mkern 1mu}. \hfill \\ \end{gathered}$

Thanks to (4.8) once more, we deduce that

$\begin{gathered} \frac{1}{2}\int_{{Q^t}} {} {\left| {q_t^\varepsilon } \right|^2}dxds + \frac{\varepsilon }{2}\int_\Omega | \nabla {q^\varepsilon }(t){|^2}dx \hfill \\ \leq c\int_{{Q^t}} {} |{q^\varepsilon }{|^2}dxds + c\int_t^T {} \left\| {\mu _t^\varepsilon (s)} \right\|_3^2\left\| {{p^\varepsilon }(s)} \right\|_V^2ds + c\int_{{Q^t}} {} |p_t^\varepsilon {|^2}dxds. \hfill \\ \end{gathered}$

Thus, (3.8) and (4.17) imply that

${\left\| {q_t^\varepsilon } \right\|_{{L^2}(0, T;H)}} + {\varepsilon ^{1/2}}{\left\| {{q^\varepsilon }} \right\|_{{L^\infty }(0, T;V)}} \leq c{\mkern 1mu}.$

(4.19)

S_TEP 4: Now, we let $\varepsilon$ tend to zero and construct a solution to (4.3)-(4.6). By (4.17)-(4.19) we have, at least for a subsequence,

$\begin{gathered} {p^\varepsilon } \to p\quad {\text{weakly star in }}{H^1}(0, T;H) \cap {L^\infty }(0, T;V) \cap {L^2}(0, T;W){\mkern 1mu}, \hfill \\ {q^\varepsilon } \to q\quad {\text{weakly in }}{H^1}(0, T;H){\mkern 1mu}, \hfill \\ \varepsilon {q^\varepsilon } \to 0\quad {\text{strongly in }}{\mkern 1mu} {L^\infty }(0, T;V), \hfill \\ \end{gathered}$

for some pair (p, q) satisfying the regularity requirements (4.9). By accounting for (3.9) and the Lipschitz continuity of g and g′, it is straightforward to see that (p, q) is a solution to problem (4.3)-(4.6). This completes the proof.

C_OROLLARY 4.3: Suppose that (A1)-(A5) are fulfilled, and assume that $\bar u \in {U_{{\text{ad}}}}$ is an optimal control of (CP) with associated state $(\bar \rho, \bar \mu)=S(\bar u)$ and adjoint state (p, q). Then it holds the variational inequality

$\int_Q {} (p + {\beta _3}\bar u)(v-\bar u)dx dt \geq {\mkern 1mu} 0\quad \forall {\mkern 1mu} v \in {U_{{\text{ad}}}}{\mkern 1mu}.$

(4.20)

P_ROOF: We fix $v \in {U_{{\text{ad}}}}$ and choose $h=v-\bar u$. Then, we write the linearized system (3.1)-(3.4) and multiply the equations (3.1) and (3.2) by p and q, respectively. At the same time, we consider the adjoint system and multiply the equations (4.3) and (4.4) by $-\eta$ and $-\xi$, respectively. Then, we add all the equalities obtained in this way and integrate over Q. Many terms cancel out. In particular, this happens for the contributions given by the Laplace operators, due to the boundary conditions (3.3) and (4.5), as well as for the terms involving $DB[\bar \rho]$ and $DB{[\bar \rho]^*}$, by the definition of adjoint operator (see (4.7)). Thus, it remains

$\begin{gathered} \int_Q {} (2g'(\bar \rho){{\bar \rho }_t}{\mkern 1mu} \eta {\mkern 1mu} p + (1 + 2g(\bar \rho)){\mkern 1mu} {\eta _t}{\mkern 1mu} p + (1 + 2g(\bar \rho)){\mkern 1mu} \eta {\mkern 1mu} {p_t})dx dt \hfill \\ + \int_Q {} ({{\bar \mu }_t}{\mkern 1mu} g'(\bar \rho){\mkern 1mu} \xi {\mkern 1mu} p + \bar \mu {\mkern 1mu} g''(\bar \rho){\mkern 1mu} {{\bar \rho }_t}{\mkern 1mu} \xi {\mkern 1mu} p + \bar \mu {\mkern 1mu} g'(\bar \rho){\mkern 1mu} {\xi _t}{\mkern 1mu} p + \bar \mu {\mkern 1mu} g'(\bar \rho){\mkern 1mu} \xi {\mkern 1mu} {p_t})dx dt \hfill \\ + \int_Q {} ({\xi _t}{\mkern 1mu} q + \xi {\mkern 1mu} {q_t})dx dt \hfill \\=\int_Q {} ((v-\bar u){\mkern 1mu} p-{\beta _2}(\bar \mu-{\mu _Q}){\mkern 1mu} \eta-{\beta _1}(\bar \rho-{\rho _Q}){\mkern 1mu} \xi)dx dt \hfill \\ \end{gathered}$

Now, we observe that the expression on the left-hand side coincides with

$\int_Q {} {\partial _t}\{ (1 + 2g(\bar \rho)){\mkern 1mu} \eta {\mkern 1mu} p + \bar \mu {\mkern 1mu} g'(\bar \rho){\mkern 1mu} \xi {\mkern 1mu} p + \xi q\} dx dt{\mkern 1mu}.$

Thus, it vanishes, due to the initial and final conditions (3.4) and (4.6). This implies that

$\int_Q {} ({\beta _1}(\bar \rho-{\rho _Q}){\mkern 1mu} \eta + {\beta _2}(\bar \mu-{\mu _Q}){\mkern 1mu} \xi)dx dt=\int_Q {} (v-\bar u){\mkern 1mu} pdx dt.$

Therefore, (4.20) follows from (3.60).

R_EMARK 4: The variational inequality (4.20) forms together with the state system (1.2)-(1.5) and the adjoint system (4.3)-(4.6) the system of first-order necessary optimality conditions for the control problem (CP). Notice that in the case ${\beta _3} > 0{\mkern 1mu}$ the function $\, -\beta_3^{-1}p\,$ is nothing but the L²(Q) orthogonal projection of ${\bar u}$ onto ${U_{{\text{ad}}}}$.

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Acknowledgements

This work received a partial support from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e loro Applicazioni) of INDAM (Istituto Nazionale di Alta Matematica) and the IMATI-C.N.R. Pavia for PC and GG.

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