Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations

1.   Introduction

In several situations the motion of incompressible electrical conducting fluid can be modeled by the magnetohydrodynamic (MHD) equations, which correspond to the Navier-Stokes equations coupled with the Maxwell equations. In the presence of a free motion of heavy ions, not directly due to the electrical field (see Schlüter [19] and Pikelner [15]), the MHD equations can be reduced to

together with the following boundary and initial conditions:

In the previous expressions, ${{\boldsymbol u}}$ and ${{\boldsymbol h}}$ are respectively the unknown velocity and magnetic field; $p^{\ast }$ is the unknown hydrostatic pressure; $w$ is an unknown function related to the heavy ions (in such a way that the density of electric current, $j_{0},$ generated by this motion satisfies the relation ${\rm rot} j_{0} = -\sigma \nabla \omega)$, $\rho$ is the density of mass of the fluid (assumed to be a positive constant); $\mu >0$ is the constant magnetic permeability of the medium; $\sigma >0$ is the constant electric conductivity; $\eta >0$ is the constant viscosity of the fluid and ${{\boldsymbol f}}$ is a given external force field.

Due to its importance, the MHD system has been discussed in a broad variety of studies encompassing subjects such as the existence of weak solutions and strong solutions, uniqueness and regularity criteria. See e.g. [6], [13], [12], [14], [7], [17], [18] and the references therein.

In the present work we discuss the stability of stationary solutions of the MHD equations in two- and three-dimensional bounded domains with respect to both initial conditions and external forcing variations. Under certain regularity hypotheses on the problem data, we establish in Theorem 3.2 the aforementioned stability in the $L^2$-norm for weak slow flow stationary solutions. Additionally, in Theorem 4.4 and Theorem 5.2 we discuss respectively the $H^1$-stability and the $H^2$-stability for strong solutions. We note that, for a fixed given external force field, our results in particular imply the asymptotic stability of such stationary solutions.

The issue of stability of solutions is an important one, since solutions of any dynamical system are thought to be physically reasonable only if they are stable. There exists a number of ways in which stability can be examined. In past years, many efforts have been made to study the asymptotic behavior of classical Navier-Stokes equations. We refer the reader to Heywood and Rannacher [11], Beirão da Veiga [2], Qu and Wang [16], Zhang [21] and the references therein.

A few of the references mentioned above, e.g. [8], are closely related to the contents of this paper. In effect, it was shown in [8] that, under condition (52) stated in Section 5 below, the strong solution of the two dimensional Navier-Stokes equation is asymptotically stable in a bounded domain of $\mathbb{R}^2$. In this paper we establish the corresponding result for the magnetohydrodynamic equations, assuming instead condition (34) below, which is weaker than the used in [8], both in two dimensional and three dimensional domains. Further, under hypotheses (52) we will show that stability actually holds in the $H^2$-norm. Thus, our results improve the existing ones even for Navier-Stokes equations.

2.   Notation and preliminaries

We will consider the usual Sobolev spaces $W^{m,q}(\Omega) = \{f\in L^q(\Omega); \|\partial^\alpha f\|_{L^q(\Omega)} < +\infty, |\alpha|\leq m\}$, for $m = 0, 1, 2, \dots, \ 1\leq q \leq +\infty$, with the usual norm. When $q = 2$, we write $H^m(\Omega) = W^{m, 2}(\Omega)$ and set $H^m_0 (\Omega) =$ closure of $C^\infty_0(\Omega)$ in $H^m(\Omega)$. The $L^q$-norm is denoted by $\|\cdot\|_{L^q(\Omega)}$. When $q = 2$, the $L^2$-norm is denoted by $\|\cdot\|$ and the associated inner product in $L^2(\Omega)$ by $(\cdot, \cdot)$.

If $X$ is a Banach space, we denote by $L^q(0,T;X)$ the Banach space of the $X$-valued functions defined in the interval $[0, T]$ that are $L^q$-integrable in the sense of Bochner. In addition, vector spaces will be denoted by boldface letters.

We also consider the following spaces of divergence free functions:

Throughout the paper, the Helmholtz projection $P$ is the orthogonal projection from ${{{\boldsymbol L}}}^2(\Omega)$ into ${{{\boldsymbol H}}}$ and $A = -P\Delta$ with $D(A) = {{{\boldsymbol V}}}\cap{{{\boldsymbol H}}}^2(\Omega)$ is the usual Stokes operator. We observe that, by the regularity of the Stokes operator, it is usually assumed that $\Omega$ is of class $C^3$ in order to apply Cattabriga's results [5]. However, we use the stronger results of Amrouche and Girault [1], which imply, in particular, that when $A{{\boldsymbol u}} \in {{{\boldsymbol L}}}^2(\Omega)$, then ${{\boldsymbol u}} \in {{{\boldsymbol H}}}^2(\Omega)$ and $\|{{\boldsymbol u}}\|_{{{\boldsymbol H}}^2(\Omega)}$ and $\|A{{\boldsymbol u}}\|$ are equivalent norms when $\Omega$ is of Class $C^{1,1}$.

For ease of reference, we also recall the following inequalities which are consequences of the Sobolev and Hölder inequalities:

Lemma 2.1. Let $\Omega \subseteq \mathbb{R}^3$ be bounded. Then

(a) There is a constant $C_L>0$ such that for any ${{\boldsymbol u}}\in {{\boldsymbol V}}$

(b) If each integral makes sense, for $\, p, q, r >0$ and $1/p + 1/q + 1/r = 1$, we have

We also need the following regularity result for the Stokes problem (see Temam [20])

Proposition 1. Let $\Omega$ be an open set in $\mathbb{R}^n$ of class $C^r$, where $n = 2$ or $3$ and $r = \max (m+2,2)$ for some integer $m\geq -1$, and let ${{\boldsymbol g}} \in {{\boldsymbol W}}^{m,q}(\Omega)$, $1< q < \infty$. Then there exist unique functions ${{\boldsymbol v}}$ and $\eta$ (to be precise, $\eta$ is unique up to a constant) that are solutions of (5) and satisfy

with

where $C$ is a constant depending on $q, \mu, m, \Omega$.

2.1.   Mathematical setting of the problem

By applying the Helmholtz operator $P$ to both sides of the first equation in problem (1), and by taking into account the previous considerations, one obtains the operational form of the problem:

Here we have set

The associated variational formulation is the following: to find $({{\boldsymbol u}},{{\boldsymbol h}})$ in suitable functional spaces such that ${{\boldsymbol u}}(0) = {{\boldsymbol u}}_0$, ${{\boldsymbol h}}(0) = {{\boldsymbol h}}_0$ and, for every $({{\boldsymbol v}},{{\boldsymbol b}}) \in {{\boldsymbol V}}\times {{\boldsymbol V}}$, the following holds:

The corresponding stationary system in operational form is

In this last system we considered a time-independent external force field ${{\boldsymbol f}}_\infty$, possibly different from the previous ${{\boldsymbol f}}$, because we want to check also the stability associated to changes in the external force field.

This last problem, in its associated variational formulation becomes: find $({{\boldsymbol u}}_\infty, {{\boldsymbol h}}_\infty) \in {{\boldsymbol V}} \times {{\boldsymbol V}}$ such that, for every $({{\boldsymbol v}},{{\boldsymbol b}}) \in {{\boldsymbol V}} \times {{\boldsymbol V}}$, the following holds:

We call such pair $({{\boldsymbol u}}_\infty, {{\boldsymbol h}}_\infty)$ a weak solution of the stationary problem (9) (or (8)).

By using the Galerkin method, it is possible to show the following result on existence of weak solutions of (9) (see Chizhonkov [6]):

Proposition 2. Problem (9) admits at least one weak solution $( {{\boldsymbol u}}_{\infty}, {{\boldsymbol h}}_{\infty}) \in {{\boldsymbol V}}\times {{\boldsymbol V}}$. Further, it satisfies the estimate

Under smallness conditions, we also have uniqueness of such solutions:

Proposition 3. (Uniqueness) Any stationary weak solution satisfying the conditions

where $0< C_L$ is the constant appearing in (3), is unique.

Proof. (of Proposition 3) Let $({{\boldsymbol u}}^1_\infty, {{\boldsymbol h}}^1_\infty)$ be a slow-flow solution of (9), that is, a weak solution satisfying (11) and (12), and let $({{\boldsymbol u}}^2_\infty, {{\boldsymbol h}}^2_\infty)$ be another tentative weak solution of (9). By setting ${{\boldsymbol u}} = {{\boldsymbol u}}^1_\infty - {{\boldsymbol u}}^2_\infty$ and ${{\boldsymbol h}} = {{\boldsymbol h}}^1_\infty - {{\boldsymbol h}}^2_\infty$, we have

We take ${{\boldsymbol v}} = {{\boldsymbol u}}$ and ${{\boldsymbol b}} = {{\boldsymbol h}}$ in the above equalities and obtain

By Lemma 2.1, we have

By adding equalities (13) and (14), using Young's inequality, and the last estimates, we obtain

which, together with hypothesis (11) and (12), implies that $\|\nabla{{\boldsymbol u}}\| = {\bf 0}$ and $\|\nabla{{\boldsymbol h}}\| = {\bf 0}$. Since $({{\boldsymbol u}}, {{\boldsymbol h}}) \in {{\boldsymbol V}} \times {{\boldsymbol V}}$, we obtain that ${{\boldsymbol u}} = {\bf 0}$ and ${{\boldsymbol h}} = {\bf 0}$, i.e., $( {{\boldsymbol u}}^1_\infty, {{\boldsymbol h}}^1_\infty) = ( {{\boldsymbol u}}^2_\infty, {{\boldsymbol h}}^2_\infty)$ which completes the proof.

Remark 1. Since, by (10), $\| {{\boldsymbol u}} \|_{{{\boldsymbol L}}^3(\Omega)} \leq C \| \nabla {{\boldsymbol u}} \|$ and $\| {{\boldsymbol h}} \|_{{{\boldsymbol L}}^3(\Omega)} \leq C \| \nabla {{\boldsymbol h}} \|$, conditions (11) and (12) can be interpreted either as saying that $\nu, \gamma$ are suficiently large or that $\Vert {{\boldsymbol f}}_\infty \Vert_{{{\boldsymbol V}}^*}$ is sufficiently small. In these cases, we say that the associated $( {{\boldsymbol u}}_{\infty}, {{\boldsymbol h}}_{\infty})$ is a stationary slow flow solution.

Next, we show that the regularity of the weak solutions of the boundary value problem (9) correlates with that of ${{\boldsymbol f}}_\infty$, i.e., the more regular ${{\boldsymbol f}}_\infty$ is, the more regular the indicated solutions will be. To this end, we note that, by putting the nonlinearities on the right-hand side of (9), the stationary problem is equivalent to the following two coupled Stokes problems. The first Stokes problem is:

where $p_\infty = \left(p_\infty^{\ast }+\frac{\mu }{2}{{\boldsymbol h}}_{\infty}^{2}\right)$. The second Stokes problem is

Proposition 4. Under the assumptions of Proposition 2 and the condition ${{\boldsymbol f}}_\infty \in {{\boldsymbol L}}^2(\Omega)$, we have ${{\boldsymbol u}}_\infty,{{\boldsymbol h}}_\infty \in {{\boldsymbol H}}^2(\Omega)\cap {{\boldsymbol V}}$. Moreover, the following inequality holds:

where $\Psi$ is a continuous and nondecreasing function of its argument such that $\Psi (0) = 0$.

Proof. We begin by discussing the first problem (15) in $({{\boldsymbol u}}_\infty,p_\infty)$ with given ${{\boldsymbol h}}_\infty \in {{\boldsymbol V}}$. By using the $L^q$-regularity properties of the Stokes problem given in Proposition 1, we conclude that

Fix $q = 3/2$ and let us estimate the terms on the right-hand side $\mathcal{F}$ of (15); by using the embedding $H^1_0(\Omega) \hookrightarrow L^6(\Omega)$ and Holder's inequality, we have

hence

Similarly,

As ${{\boldsymbol f}}_\infty \in {{\boldsymbol L}}^2(\Omega)$, we have $\alpha{{\boldsymbol f}}_\infty -\alpha({{\boldsymbol u}}_\infty\cdot\nabla){{\boldsymbol u}}_\infty +( {{\boldsymbol h}}_\infty\cdot\nabla){{\boldsymbol h}}_\infty \in {{\boldsymbol L}}^{3/2}(\Omega)$, so that $\mathcal{F} \in {{\boldsymbol L}}^{3/2}(\Omega)$. By (18), we conclude that ${{\boldsymbol u}}_\infty \in {{\boldsymbol W}}^{2,3/2}(\Omega)$ and $p_\infty \in W^{1,3/2}(\Omega)$.

Next, we consider the problem (16) and use the already known fact that ${{\boldsymbol u}}_\infty\in {{\boldsymbol W}}^{2,3/2}(\Omega)$. Again by the regularity properties of the Stokes problem, we have

We must now estimate the terms on the right-hand side $\mathcal{G}$ of (16). We have:

hence

Similarly,

Consequently, $\mathcal{G} \in {{\boldsymbol L}}^{3/2}(\Omega)$, and so ${{\boldsymbol h}}_\infty \in {{\boldsymbol W}}^{2,3/2}(\Omega)$ and $\omega_\infty \in W^{1,3/2}(\Omega)$.

Now, since $W^{2,3/2}(\Omega) \hookrightarrow W^{1,3}(\Omega) \hookrightarrow L^6(\Omega)$, we obtain

and

Using the above and noting that $\alpha{{\boldsymbol f}}_\infty \in{{\boldsymbol L}}^2(\Omega)$, we obtain $\mathcal{F} \in {{\boldsymbol L}}^2(\Omega)$. Thus, (18) with $q = 2$ yields ${{\boldsymbol u}}_\infty\in{{\boldsymbol H}}^2(\Omega)$ and $p_\infty\in H^1(\Omega)$.

We conclude that

and

Consequently, $\mathcal{G} \in {{\boldsymbol L}}^2(\Omega)$. Thus, from (21) with $q = 2$, we obtain ${{\boldsymbol h}}_\infty\in{{\boldsymbol H}}^2(\Omega)$ and $\omega_\infty \in {{\boldsymbol H}}^1(\Omega)$.

By following the previous estimates and using Proposition 2, we obtain (17), which completes the proof.

In the remainder of this work, $( {{\boldsymbol u}}_{\infty}, {{\boldsymbol h}}_{\infty})$ will denote a stationary slow-flow solution of the type discussed in this Section, i.e., $( {{\boldsymbol u}}_{\infty}, {{\boldsymbol h}}_{\infty})$ satisfies the conclusions of Proposition 2 and 3.

3.   ${{\boldsymbol L}}^2$-stability

Definition 3.1. For any given ${{\boldsymbol u}}_0, {{\boldsymbol h}}_0 \in {{\boldsymbol H}}$ and ${{\boldsymbol f}} \in L^\infty(0,\infty, {{\boldsymbol V}}^*)$, we say that a pair $({{\boldsymbol u}}, {{\boldsymbol h}})$ is a weak solution of (1)-(2) if ${{\boldsymbol u}},{{\boldsymbol h}} \in L^\infty(0,\infty, {{\boldsymbol H}}) \cap L^2_{loc} (0,\infty, {{\boldsymbol V}})$ and (7) holds.

Proposition 5. Let ${{\boldsymbol f}}\in L^\infty(0,\infty;{{\boldsymbol V}}^*)$, ${{\boldsymbol f}}_\infty \in {{\boldsymbol V}}^*$ and let $({{\boldsymbol u}}, {{\boldsymbol h}})$ be a weak solution of (1)-(2) such that

Further, we assume that $({{\boldsymbol u}}_\infty,{{\boldsymbol h}}_\infty )$ is a weak slow- flow solution of (9), i.e., (11) and (12) hold. Then there exists a positive constant $\beta_0 >0$ such that, for every $\beta \in (0, \beta_0]$, we have

Proof. Let

Then

By taking ${{\boldsymbol v}} = {{\boldsymbol w}}$ in (25), we obtain

On the other hand, taking ${{\boldsymbol b}} = {{\boldsymbol z}}$ in (26), we obtain

To estimate the terms on the right-hand side of the above expressions, we first note that

Similarly,

The above equalities together with (27) and (28) imply the following differential identity

The terms on the right-hand side of (29) can be estimated as follows. From Lemma 2.1 and the Young inequality, we have

Similarly,

Using the above in equality (29), we obtain

Now we observe that (11) and (12) imply that

Therefore, by (30), we have

Now, using the embedding ${{\boldsymbol H}}^1_0(\Omega) \hookrightarrow {{\boldsymbol L}}^2(\Omega)$, we have

for every $\beta \in (0, \beta_0]$, where

with $C_{e1}$ equal to the embedding constant of ${{\boldsymbol H}}^1_0(\Omega) \hookrightarrow {{\boldsymbol L}}^2(\Omega)$. By integrating this inequality, we obtain the desired decay property (23).

In the next section we will also need the following estimate.

Proposition 6. Let $({{\boldsymbol u}}, {{\boldsymbol h}})$, $({{\boldsymbol u}}_\infty, {{\boldsymbol h}}_\infty)$ and $\beta > 0$ be as in Proposition 5. Then,

Proof. Multiplying inequality (31) by $e^{\beta t}$ yields

Now, integrating this last inequality from $0$ to $t$ and then multiplying by $e^{-\beta t}$, we obtain

Using (23) and Fubini's theorem, the above yields

whence estimate (33) follows.

We can now prove the following stability result.

Theorem 3.2. ($L^2$-stability) Assume the hypotheses of Proposition 5 and also that $\lim_{t\rightarrow \infty} \Vert {{\boldsymbol f}}(t) - {{\boldsymbol f}}_\infty\Vert_{{{\boldsymbol V}}^*} = 0$. Then $\Vert {{\boldsymbol u}}(t)-{{\boldsymbol u}}_\infty\Vert \rightarrow 0$ and $\Vert {{\boldsymbol h}}(t)-{{\boldsymbol h}}_\infty\Vert \rightarrow 0$ as $t \rightarrow +\infty$.

Proof. We must show that $\lim_{t \rightarrow +\infty} \alpha \Vert {{\boldsymbol u}}(t) -{{\boldsymbol u}}_\infty \Vert^2 + \Vert {{\boldsymbol h}}(t) - {{\boldsymbol h}}_\infty \Vert^2 = 0$, that is, given any $\epsilon > 0$ ($<1$, without loss of generality), there exists a $T_{\epsilon}$ such that $\alpha \Vert {{\boldsymbol u}}(t) -{{\boldsymbol u}}_\infty \Vert^2 + \Vert {{\boldsymbol h}}(t) - {{\boldsymbol h}}_\infty \Vert^2 < \epsilon$ for $t > T_{\epsilon}$.

To find a $T_{\epsilon}$, let us start by considering $\delta > 0$ (which will be chosen later in function of $\epsilon$). Since $\lim_{t\rightarrow \infty} \Vert {{\boldsymbol f}}(t) - {{\boldsymbol f}}_\infty\Vert_{{{\boldsymbol V}}^*} = 0$, we can choose a $T_\delta$ such that $\Vert {{\boldsymbol f}}(t) - {{\boldsymbol f}}_\infty\Vert_{{{\boldsymbol V}}^*} < \delta$ for $t > T_\delta$. Also note that $\Vert {{\boldsymbol f}}(t) -{{\boldsymbol f}}_\infty\Vert_{{{\boldsymbol V}}^*} \leq \Vert {{\boldsymbol f}} \|_{L^\infty (0,\infty; {{\boldsymbol V}}^*)} + \| {{\boldsymbol f}}_\infty\Vert_{{{\boldsymbol V}}^*}$ for all $t$.

Now, by (23) with a fixed $\beta \in (0, \beta_0]$, we have

We now choose $\delta$ so that $\delta ^2 \alpha^2 / \beta < \epsilon/3$, i.e., $\delta < (\beta \epsilon/(3\alpha^2))^{1/2}$, which yields the corresponding $T_{\delta}$. Then, from the last estimate we see that it suffices to choose $T_\epsilon$ sufficiently large so that, for $t > T_\epsilon$, we have

These conditions are satisfied with

Thus, $\alpha \Vert {{\boldsymbol u}}(t) - {{\boldsymbol u}}_\infty \Vert^2 + \Vert {{\boldsymbol h}}(t)- {{\boldsymbol h}}_\infty \Vert^2 < \epsilon$, which completes the proof.

Remark 2. When $n = 2$, there exists a unique global weak solution $({{\boldsymbol u}}, {{\boldsymbol h}} )$ of (7) satisfying the initial condition $({{\boldsymbol u}}_0,{{\boldsymbol h}}_0) \in {{\boldsymbol H}}\times{{\boldsymbol H}}$. Moreover, it is not difficult to check that ${{\boldsymbol u}}_t, {{\boldsymbol h}}_t \in L^2(0,T;{{\boldsymbol V}}^*)$ (see [17]). Thus the estimates stated in Proposition 5 and therefore the conclusions of Theorem 3.2, hold. In particular, the above implies that any stationary slow-flow solution is weakly asymptotically stable.

When $n = 3$ and ${{\boldsymbol u}},{{\boldsymbol h}} \in L^s(0,T;{{\boldsymbol L}}^r(\Omega))$ with $2/s + 3/r \leq 1$ and $r>3$, it can be shown that the solution satisfies ${{\boldsymbol u}}_t, {{\boldsymbol h}}_t \in L^2(0,T;{{\boldsymbol V}}^*)$ and is, furthermore, unique (see [7]). In this setting, the estimates stated in Proposition 5 and therefore the conclusions of Theorem 3.2 hold. Moreover, as before, any stationary slow-flow solution is weakly asymptotically stable.

The following is an immediate corollary of the theorem

Corollary 1. If we set ${{\boldsymbol f}}(t) = {{\boldsymbol f}}_\infty$ in the Theorem 3.2, then the convergence rate there is exponential in the $L^2$-norm.

4.   ${{\boldsymbol H}}^1$-stability

Definition 4.1. Let ${{\boldsymbol u}}_0, {{\boldsymbol h}}_0 \in {{\boldsymbol V}}$ and let ${{\boldsymbol f}} \in L^\infty([0,\infty); {{\boldsymbol L}}^2(\Omega))$. By a strong solution of problem (1) we mean a pair of vector-valued functions ${{\boldsymbol u}},{{\boldsymbol h}}$ such that ${{\boldsymbol u}},{{\boldsymbol h}} \in L^\infty(0,\infty;{{\boldsymbol V}}) \cap L^2_{loc}(0,\infty;{{\boldsymbol H}}^2(\Omega)\cap {{\boldsymbol V}})$ which satisfy (1).

The following conditions on the initial data will remain in force throughout this section:

Under assumptions (34), the existence and uniqueness of a local solution was established in [4], as follows:

Theorem 4.2. The conditions (34) imply the existence of a positive constant $T>0$ and functions ${{\boldsymbol u}},{{\boldsymbol h}} \in C([0,T);V)$ and $p, \omega \in L^2(0,T;H^1(\Omega)\backslash \mathbb{R})$ which are the unique strong solution of (1)-(2).

The following global existence theorem was proved in [18]:

Theorem 4.3. Assume that $n = 2$ or that $n = 3$ and the constant $M_1$ in (34) is appropriately small. Then the solution in Theorem 4.2 exists for every $t\geq 0$ and it satisfies

As in the case of the standard Navier-Stokes equations, it is unknown whether or not the conclusion of Theorem 4.3 holds in general for large data in three dimensions. In what follows, we will work under the assumption that it does, i.e., we henceforth assume that there exist constants $M_2$ and $T$, where $0<T \leq \infty$ is as in Theorem 4.2, such that

We note that it is also possible to carry out the following discussion without making the above assumption, namely by repeating the preceding smallness condition in three dimensions whenever needed. This approach, however, complicates the exposition and we avoid it.

Clearly, by Theorem 4.3, condition (35) holds without additional hypotheses in the two-dimensional case.

Remark 3. Assumption (35) was previously used by Heywood [9] and Heywood and Rannacher [11] in the study of convergence of Galerkin and finite element methods in the study of the classical Navier-Stokes equations, respectively.

Proposition 7. Let $({{\boldsymbol u}}_\infty,{{\boldsymbol h}}_\infty)$ and $({{\boldsymbol u}},{{\boldsymbol h}})$ be as in the last Section. Assume also that (34), (35) hold (thus $({{\boldsymbol u}}_\infty,{{\boldsymbol h}}_\infty)$ and $({{\boldsymbol u}},{{\boldsymbol h}})$ are strong solutions). Then, there exists a positive constant $\kappa_0$, which depends only on $\Omega$ and on given parameters of the problem, such that, for every $\kappa \in (0,\kappa_0]$ we have:

Proof. By taking ${{\boldsymbol v}} = A{{\boldsymbol w}}$ in (25) and ${{\boldsymbol b}} = A{{\boldsymbol z}}$ in (26), we obtain

Next, we estimate the terms on the right-hand sides:

Consequently

Now we observe that

for every $\kappa \in (0, \kappa_0]$, where

Above $C_{e2}$ denotes the embedding constant of ${{\boldsymbol H}}^2(\Omega) \cap {{\boldsymbol H}}^1_0(\Omega) \hookrightarrow {{\boldsymbol H}}^1_0(\Omega)$ and $\beta_0$ is given by (32).

Thus (39) implies

Now, integrating (40) from $0$ to $t$, we have

We also note that

and, using (10) and (35), we obtain

Similarly,

Thus

and

Using the above in (41), we conclude that

Finally, the second and third terms on the right-hand side of the last inequality can be estimated using (33), which yields the estimate (36).

In the next section we will need the following estimates.

Proposition 8. Let $({{\boldsymbol u}}, {{\boldsymbol h}})$, $({{\boldsymbol u}}_\infty, {{\boldsymbol h}}_\infty)$ and $\kappa > 0$ be as in Proposition 7. Then the following inequalities hold:

and

Proof. By (39), we have

Now, integrating the above with respect to time from $0$ to $t$, we obtain

Using (33) above with $\beta = \kappa$ and $C_M>0$ depending on $M_3, M_4$, we obtain the required estimate (42).

Next, from (25) and (26), we obtain

and

These expressions imply that

Next, using (35), the embeddings $L^r(\Omega) \hookrightarrow H^1(\Omega)$, where $r = 3$ or $6$, $H^2(\Omega) \hookrightarrow L^\infty(\Omega)$ and the equivalence of norms $\Vert {{\boldsymbol v}}\Vert_{{{\boldsymbol H}}^2(\Omega)}$ and $\Vert A{{\boldsymbol u}}\Vert$, (48) yields

Similarly, from (49) we obtain

Consequently,

Using (42), the above yields the required estimate (43).

Estimate (44) is obtained similarly, using (49).

Finally, using estimate (36) and arguing exactly as in the proof of Theorem 3.2, we can prove the following stability result.

Theorem 4.4. (${{\boldsymbol H}}^1$-stability) Assume that $\lim_{t\rightarrow \infty} \Vert {{\boldsymbol f}}(t) - {{\boldsymbol f}}_\infty\Vert = 0$ and that (34) and (35) hold. Then $\Vert {{\boldsymbol u}}(t)-{{\boldsymbol u}}_\infty\Vert_{{{\boldsymbol H}}^1_0 (\Omega)} \rightarrow 0$ and $\Vert {{\boldsymbol h}}(t)-{{\boldsymbol h}}_\infty\Vert_{{{\boldsymbol H}}^1_0 (\Omega)} \rightarrow 0$ as $t \rightarrow +\infty$.

The following is an immediate corollary of the theorem.

Corollary 2. If in Theorem 4.4, we set ${{\boldsymbol f}}(t) = {{\boldsymbol f}}_\infty$, then the convergence rate there is exponential in the $H^1$-norm.

5.   $H^2$-stability

If one assumes greater regularity of the initial data, stability in the $H^2$-norm is attained. To show this, we assume throughout this section that

The following result was proved in [18].

Theorem 5.1. Let the hypotheses be as in Theorem 4.3 and assume, in addition, that (52) holds. Then ${{\boldsymbol u}}, {{\boldsymbol h}} \in C([0,\infty);{{\boldsymbol V}}\cap{{\boldsymbol H}}^2(\Omega)) \cap C^1 ([0,\infty);{{\boldsymbol H}})$, where ${{\boldsymbol u}}$ and ${{\boldsymbol h}}$ are as in Theorem 4.3. Further, there exists a finite positive constant $C$ such that

Using the preceding statements, we obtain the following estimates:

Proposition 9. Assume that (35), (52) and the uniqueness condition in the stationary system hold and let $({{\boldsymbol u}},{{\boldsymbol h}})$ be a strong solution as in Theorem 5.1. Then there exists a positive constant $\tilde{\beta}_0 >0$ such that, for every $\tilde{\beta} \in (0, \tilde{\beta}_0]$, the following holds

Here, $\alpha \Vert {{\boldsymbol w}}_t (0) \Vert^2 \leq C\Vert A {{\boldsymbol w}} (0) \Vert^2 + C \Vert A{{\boldsymbol z}} (0) \Vert^2 + C\Vert {{\boldsymbol f}} (0) -{{\boldsymbol f}}_\infty \Vert^2$ and $\Vert {{\boldsymbol z}}_t (0) \Vert^2 \leq C\Vert A {{\boldsymbol w}} (0) \Vert^2 + C \Vert A{{\boldsymbol z}} (0) \Vert^2$.

Proof. We differentiate (25) and (26) with respect to $t$ and set ${{\boldsymbol v}} = {{\boldsymbol w}}_t$ and ${{\boldsymbol b}} = {{\boldsymbol z}}_t$ there. Noting that ${{\boldsymbol w}}_t = {{\boldsymbol u}}_t$ and ${{\boldsymbol z}}_t = {{\boldsymbol h}}_t$, we conclude that

and

Next, we estimate the terms on the right-hand sides:

Now, by adding the equalities (56) and (57) and using the last obtained estimates, we get

where

are bounded functions.

By working similarly with (58), whose left-hand is analogous to the corresponding one of (31), we obtain (55) for $\tilde{\beta} \in (0, \tilde{\beta}_0]$ with

where as before $C_{e1}$ is the embedding constant of ${{\boldsymbol H}}^1_0(\Omega) \hookrightarrow {{\boldsymbol L}}^2(\Omega)$.

Finally, the stated estimates for $\alpha \Vert {{\boldsymbol w}}_t (0)\Vert$ and $\Vert {{\boldsymbol z}}_t(0) \Vert$ are consequences of (50) and (51), respectively.

Next, we have

Proposition 10. Let the hypotheses be as in Proposition 9. Then, there exists a positive constant $\tilde{\kappa}_0$, which only depends $\Omega$ and on the given parameters of the problem, such that, for every $\tilde{\kappa} \in (0, \tilde{\kappa}_0]$, we have:

and

Proof. Let $\kappa_0$ and $\tilde{\beta}_0$ be the positive constants in Propositions 7 and 9, respectively, and set

It follows from (46) that

Now, taking $\tilde{\kappa} \leq \tilde{\kappa}_0$ and using (55) together with the hypotheses on $({{\boldsymbol u}}_\infty,{{\boldsymbol h}}_\infty)$ and $({{\boldsymbol u}},{{\boldsymbol h}})$ in this last inequality, we obtain

Using the estimates for $\alpha \Vert {{\boldsymbol w}}_t (0)\Vert$ and $\alpha \Vert {{\boldsymbol z}}_t (0)\Vert$ given in (43), (44) and Proposition 9, we obtain estimate (59) from the preceding inequality.

Next, (47) implies that

Arguing as before, the above inequality yields estimate (60).

Finally, arguing exactly as in the proof of Theorem 3.2, estimates (59) and (60) with a fixed $\tilde{\kappa} \in (0, \tilde{\kappa}_0]$ yield following stability result.

Theorem 5.2. Let the hypotheses be as in Proposition 10 and assume, in addition, that $\lim_{t\rightarrow \infty} \Vert {{\boldsymbol f}}(t) - {{\boldsymbol f}}_\infty\Vert = 0$ and $\lim_{t \rightarrow \infty} \Vert {{\boldsymbol f}}_t(t) \Vert = 0$. Then $\Vert {{\boldsymbol u}}(t)-{{\boldsymbol u}}_\infty\Vert_{{{\boldsymbol H}}^2(\Omega)} \rightarrow 0$ and $\Vert {{\boldsymbol h}}(t)-{{\boldsymbol h}}_\infty\Vert_{{{\boldsymbol H}}^2(\Omega)} \rightarrow 0$ as $t \rightarrow +\infty$.

The following is an immediate corollary of the theorem.

Corollary 3. If we set ${{\boldsymbol f}}(t) = {{\boldsymbol f}}_\infty$ in Theorem 5.2, then the convergence rate there is exponential in the $H^2$-norm.

Acknowledgments

J.L. Boldrini was partially supported by CNPq (Brazil) Grant 306182/2014-9. J. Bravo-Olivares and E. Notte-Cuello were partially supported by project DIDULS-PTE16151, Universidad de La Serena. M.A. Rojas-Medar was partially supported by CAPES-PRINT 88887.311962/2018-00 (Brazil) and Project UTA-Mayor, 4753-20, Universidad de Tarapacá (Chile).