Stability of stationary solutions to outflow problem for compressible viscoelastic system in one dimensional half space

1.   Introduction

This paper studies the compressible viscoelastic system in the one-dimensional half space $\mathbb{R}_+ = (0, \infty)$:

Here $\rho = \rho(t, x)$, $v = v(t, x)$, and $F = F(t, x)$ are the unknown density, velocity field, and deformation tensor, respectively, at time $t\geq 0$ and position $x\in \mathbb{R}_+$; $P(\rho)$ stands for the pressure assumed to be a smooth function of $\rho$ satisfying $P^\prime(\rho) > 0$ and $P^{\prime\prime}(\rho) > 0$ for $\rho > 0$; $\nu > 0$ is the viscosity coefficient; $\beta > 0$ is the strength of the elasticity. In particular, if we set $\beta = 0$, the systems (1.1) and (1.2) become the usual compressible Navier-Stokes equation.

We impose the initial condition and boundary conditions at $x = \infty$ and $x = 0$:

Here, the end states $\rho_+, v_+$ and $F_+$ are given constants with $\rho_+ > 0$, and $v_b$ is a given constant assumed to be $v_b < 0$ for considering the situation that the fluid flows out from the boundary $x = 0$. Throughout this paper, we consider the initial boundary problems (1.1)–(1.6), called the outflow problem. The aim of this paper is to show the existence and asymptotic stability of stationary solutions for the outflow problems (1.1)–(1.6) and clarify the interaction between the effect of the elastic force $\beta^2(\rho F^2)_x$ and outflow boundary condition (1.6).

We first discuss the existence and properties for the stationary solution $(\tilde{\rho}, \tilde{v}, \tilde{F})(x)$, called the boundary layer solution, solving the system:

with the conditions:

It is shown that the solution of the problems (1.7)–(1.11) exists uniquely if and only if $M_\beta\ge 1$ and $v_b < v_*$ hold. Here, $M_\beta$ is the modified Mach number given by $M_\beta: = |v_+|/\sqrt{c_+^2+\beta^2 F_+^2}$, where $c_+: = \sqrt{P^\prime(\rho_+)}$ stands for sound speed; $v_*$ is a certain negative constant determined in Section 2. In addition, the solution satisfies the following estimate:

where $\delta$ denotes $\delta: = |v_b-v_+|$. We call $(\tilde{\rho}, \tilde{v}, \tilde{F})(x)$ the non-degenerate stationary solution tending to the end state exponentially when $M_\beta > 1$, while we say $(\tilde{\rho}, \tilde{v}, \tilde{F})(x)$ by the degenerate stationary solution converging to the end state algebraically when $M_\beta = 1$. We also note that if we take $\beta$ large so that $M_\beta < 1$ under the fixed end state, then the stationary solution does not exist. This means that the stationary outflow does not occur due to the recoiling effect of strong elastic force.

We next establish the asymptotic stability of the stationary solution under the small initial perturbation $(\rho_0, v_0, F_0)-(\tilde{\rho}, \tilde{v}, \tilde{F})$, provided that $\delta$ is sufficiently small. This follows from the local-in-time solvability of (1.1)–(1.6) and the a priori estimates for the perturbation in $H^1(\mathbb{R}_+)$. Since the systems (1.1)–(1.3) is classified by a quasilinear parabolic-hyperbolic system, the local-in-time solvability is shown by the iteration method and theory of weak solutions to linear transport equations and parabolic equations, inspired by Kagei and Kawashima's paper ^[11].

To derive the a priori estimate, we need to deal with the term $\beta^2(\rho F^2-\tilde{\rho}\tilde{F}^2)_x$ when we consider the problem for a usual perturbation $(\rho, v, F)-(\tilde{\rho}, \tilde{v}, \tilde{F})$. However, it seems to be difficult to control $\rho F-\rho_+ F_+$ appearing from this term. To overcome this difficulty, we assume the following condition for $(\rho_0, F_0)$:

for $x \in \mathbb{R}_+$. We then see from (1.1), (1.3), (1.6) and (1.12) that $\rho F = \rho_+ F_+$ holds for $t\geq 0$ and $x\geq 0$. This constraint is a one-dimensional version of the following equality

which is equivalent to the conservation law of mass in the Lagrange coordinate. Here, $n\ge2$; ${\bf F}_+$ stands for given $n\times n$ matrix-valued constant; ${\bf F}(t, x) = (F^{j, k}(t, x))_{1\le j, k \le n}$ denotes an $n\times n$ matrix-valued deformation tensor, respectively.

Therefore, rewriting $F = \rho_+ F_+\rho^{-1}$, the problems (1.1)–(1.6) for $(\rho, v, F)$ is reduced to the system for $(\rho, v)$:

with the initial condition and boundary conditions at $x = \infty$ and $x = 0$:

We then prepare the reduced perturbation $(\phi, \psi): = (\rho-\tilde{\rho}, v-\tilde{v})$ and carry out its estimate as two steps. For $(\phi, \psi)$ itself, based on the idea of Kawashima, Nishibata and Zhu's paper ^[13] and the fact that $P(\rho)-\beta^2(\rho_+ F_+)^2/\rho$ monotonically increases for $\rho > 0$, we construct a suitable energy form equivalent to $|(\phi, \psi)|^2$, and use its equation and the properties of the stationary solution. The effect of the term $\beta^2(\rho_+ F_+)^2(1/\rho)_x$ is mainly involved to the proof of this estimate. Indeed, the convexity of $P(\rho)-\beta^2(\rho_+ F_+)^2/\rho$ around $\rho = \rho_+$ requires for the degenerate case $M_\beta = 1$. In order to get this convexity, we need to add the condition for the Mach number $1 < M_+ < \sqrt{\rho_+(P^{\prime\prime}(\rho_+)/2P^\prime(\rho_+))+1}$, where $M_+: = |v_+|/c_+$. For the $1$-st order spatial derivative of $(\phi, \psi)$, we utilize the structure of (1.13) and (1.14) and monotonicity of $P(\rho)-\beta^2(\rho_+ F_+)^2/\rho$ for $\rho > 0$.

We remark the reason why we only assume the regularity condition $(\rho_0-\tilde{\rho}, v_0-\tilde{v}, F_0-\tilde{F})\in H^1(\mathbb{R}_+)$ in contrast of the case $\beta = 0$ in ^[13]. In the argument in ^[13], we need the equation

to construct the a priori estimate for $\phi_x$. This means that we need to take care of $\phi_{xx}(t, x)$ and $|\phi_x(t, 0)|$. The equation (1.18) is obtained by the integration by parts with (1.6) and makes sense under $\psi \in C([0, T]; H^1(\mathbb{R}_+))$ and $\phi\in C([0, T]; H^k(\mathbb{R}_+))$ for $T > 0$ and $k\geq 2$ because $H^k(\mathbb{R}_+)\subset C^1([0, \infty))$ holds for $k\ge2$. In our setting, the equation (1.18) is not valid since we restrict the situation that $\phi\in C([0, T]; H^1(\mathbb{R}_+))$ which is not sufficient to define $\phi_{xx}(t, x)$ and $\phi_x(t, 0)$. To deal with this difficulty, Kawashima, Nishibata and Zhu in ^[13] additionally assumed that the initial perturbation belongs to the Hölder space for guaranteeing $\phi_x(t, 0)$, and then applied the method of difference quotient to avoid appearing higher derivatives such as $\phi_{xx}(t, x)$. Later, Kagei and Kawashima introduced the weak forms of the parabolic equation and first-order transport equation to show the local-in-time existence of the quasilinear parabolic-hyperbolic system in ^[11], and then used their theory to obtain the estimates for the higher order derivatives without the Hölder regularity of initial perturbation in ^[12]. Therefore, inspired the idea of ^[12], we can conclude that it is enough to suppose that initial perturbation is small only in $H^1(\mathbb{R}_+)$ to show the asymptotic stability result.

Known results. The systems (1.1)–(1.3) describing the motion of compressible viscoelastic fluid is governed in the macroscopic scale by the variational modeling. Indeed the second equation (1.2) is treated as the conservation law of momentum following from the energy dissipation law with the free energy induced by elastic solids. Here, the free energy is taken as the derivative of $\beta^2 \rho W'(F)F = \beta^2\rho F^2$, where $W(F) = \beta^2F^2/2$ denotes linear isentropic elasticity. The other equations (1.1) and (1.3) are the kinematic assumptions for $\rho$ and $F$. For more physical detail, we refer to ^{[3,4,5,17,25]}. Starting with Sideris and Thomases ^[25], the mathematical analysis of the systems (1.1)–(1.3) has been progressed mainly on the stability of the trivial motionless state. In fact, its stability is investigated by ^{[1,7,8,16,24]} in the three dimensional whole space and is studied by ^[2,9,23] in the three-dimensional bounded domain with smooth boundary case under the Dirichlet boundary condition. For the stability of non-trivial flows with non-zero velocity, the dynamics of solutions around them becomes more complicated than the trivial motionless case since the advection terms in (1.1)–(1.3) produce the additional hyperbolic aspect. Therefore, comparing to the the trivial motionless case, there are few results on their stability as follows. Ishigaki ^[10] and Haruki and Ishigaki ^[6] investigated the stability of parallel flows in the three dimensional layer, and Morando, Trakhinin and Trebeschi ^[19] and Trakhinin ^[26] studied the stability of shock waves in the two-dimensional whole space without the viscous effect.

We next review the mathematical analysis of outflow problem. For the case $\beta = 0$, it is natural to expect that the behavior of the solution in the half space is closely related to the boundary and the end states $v_b$, $v_+$ and $\rho_+$. Then, Matsumura ^[18] suggested that the long time asymptotic states will be composed of the rarefaction wave, the viscous shock wave and the boundary layer solution. The stability of stationary solution is related to the case that the asymptotic state is given by the boundary layer only. Kawashima, Nishibata and Zhu ^[13] characterized the existence of stationary solutions by determining suitable conditions for $v_b$, $v_+$ and $\rho_+$, and then showed its asymptotic stability by the energy method in the Eulerian coordinate. Nakamura, Nishibata and Yuge ^[21] established the convergence rate toward the stationary solutions as $t\to\infty$ under the small initial perturbation belonging to the weighted $L^2$ Sobolev space. Later that, Nakamura, Ueda and Kawashima ^[22] refined the convergence rate toward the degenerate stationary solution discussed in ^[13,21]. Furthermore, Kawashima and Kagei ^[11], and Nakamura and Nishibata ^[20] extended these stability results to the multidimensional case. On the other hand, if $v_b$, $v_+$ and $\rho_+$ do not satisfy the conditions for the existence of stationary solutions, the asymptotic state of the solution becomes different from the boundary layer solution. For the details, we refer to ^[14,15,27] when the rarefaction wave involves its time asymptotic state. Concerning the case $\beta > 0$, as far as the authors know, it remains open.

Outline of this paper. This paper is organized as four sections and one appendix. In Section 2, notations of several function spaces and lemmata are explained. In Section 3, we give the detailed necessary conditions for the existence of a stationary solution and provide its properties. In Section 4, the detail of the main result in this paper is stated. In Section 5, we show the asymptotic stability of the stationary solutions. In Appendix A, we prove the local-in-time existence around the stationary solutions.

2.   Notations

In this section, we introduce several function spaces and important lemmata.

For $1\leq p\leq\infty$, the symbol $L^p$ stands for the usual Lebesgue space on $\mathbb{R}_+$, and its norm is denoted by $\|\cdot\|_{L^p}$. For a non-negative integer $m\geq0$, we define $H^{m}$ as the $m$-th order $L^2$ Sobolev space on $\mathbb{R}_+$, and its norm is denoted by $\|\cdot\|_{H^m}$. For simplicity, we write $L^p = L^p\times L^p \times L^p$ (resp., $H^m = H^m\times H^m \times H^m$). The symbol $C_0^1(\mathbb{R}_+)$ denotes the set of all $C^1(\mathbb{R}_+)$ functions whose supports are compact in $\mathbb{R}_+$. We call $H_0^1(\mathbb{R}_+)$ the completion of $C_0^1(\mathbb{R}_+)$ in $H^1(\mathbb{R}_+)$. For $T > 0$, we define $C_0^1((0, T)\times\mathbb{R}_+)$ (resp., $C_0^1([0, T)\times\mathbb{R}_+)$) as the set of all $C^1((0, T)\times\mathbb{R}_+)$ (resp., $C^1([0, T)\times\mathbb{R}_+)$) functions whose supports are compact in $(0, T)\times\mathbb{R}_+$ (resp., $[0, T)\times\mathbb{R}_+$).

For $0\leq a < b\leq \infty$, a Banach space $\mathcal{X}$ endowed with norm $\|\cdot\|_{\mathcal{X}}$ and a non-negative integer $k$, we define $C^k([a, b]; \mathcal{X})$ that

where

Here, we identify $[a, \infty]: = [a, \infty)$ in the case $b = \infty$. For simplicity, we write $C([a, b]; \mathcal{X}): = C^0([a, b]; \mathcal{X})$.

For $0\leq a < b\leq \infty$, a Banach space $\mathcal{X}$ endowed with norm $\|\cdot\|_{\mathcal{X}}$ and a non-negative integer $k$, $L^2(a, b; \mathcal{X})$ and $H^k(a, b; \mathcal{X})$ denote

where

and

where

For a real number $\alpha$, we set $[\alpha]$ as its integer part. Throughout this paper, we simply regard the letters $c$, $C$, $\tilde{c}$ and $\tilde{C}$ as positive various constants.

We next state several lemmata to establish our asymptotic stability result. We first introduce the well-known Gagliardo–Nirenberg inequality to obtain a priori estimate in $H^1(\mathbb{R}_+)$.

Lemma 2.1. Let $f\in H^1(\mathbb{R}_+)$. Then, $f \in C([0, \infty))$ and it satisfies

To show the asymptotic stability of the stationary solution, the following lemma plays a role.

Lemma 2.2. [12, Lemma 4.5.] Let $T > 0$ be an arbitrary number, and $f = f(t, x)$ be $f\in C([0, T]; H_0^1(\mathbb{R}_+))\cap L^2(0, T; H^2(\mathbb{R}_+)) \cap H^1(0, T; L^2(\mathbb{R}_+))$. Then $f$ satisfies

for $0\leq t_1\leq t_2\leq T$.

We give definitions of weak solutions to the linear transport equation and the parabolic equation for studying the local-in-time existence of the unique solution to (1.1)–(1.6) around the stationary solution.

For $0\leq a < b\leq \infty$ and $k = 0, 1$, we define the function spaces $X^k(a, b)$ and $Y^k(a, b)$ as

respectively. Here, $\widetilde{H}^{k}(\mathbb{R}_+)$ is given by $\widetilde{H}^{k}(\mathbb{R}_+) : = H^{k}(\mathbb{R}_+)\cap H_0^{1}(\mathbb{R}_+)$ when $k = 1, 2$, and $\widetilde{H}^{0}(\mathbb{R}_+): = L^2(\mathbb{R}_+)$. We set the corresponding norms such that

We introduce the function space $Z^k(a, b)$ as

and the norms of $u = (\phi, \psi, \zeta)$ in $Z^k(a, b)$ are given by

For simplicity, we write $X^k(T): = X^k(0, T)$, $Y^k(T): = Y^k(0, T)$ and $Z^k(T): = Z^k(0, T)$ for $T > 0$. We also set

for $M > 0$.

We define weak solutions to the linear transport equation and the parabolic equation associated with the stationary solution of (1.1)–(1.6). The stationary solution $(\tilde{\rho}, \tilde{v}, \tilde{F})$ is the smooth solution to (1.7)–(1.11), and the existence of solutions will be discussed in Section 3. Using this stationary solution, we introduce functions $a(\psi) : = \tilde{v}+\psi$ and $b(\phi): = \tilde{\rho} + \phi$, as well as a linear operator $B:H_0^1(\mathbb{R}_+)\to H^{-1}(\mathbb{R}_+)$ defined by $\langle B \psi, \varphi \rangle : = \nu(\psi_{x}, \varphi_x)_{L^2}$ for $\psi, \varphi \in H_0^1(\mathbb{R}_+)$. Here, $H^{-1}(\mathbb{R}_+)$ is the dual space of $H_0^1(\mathbb{R}_+)$, $\langle \cdot, \cdot \rangle$ that stands for the pairing between $H^{-1}(\mathbb{R}_+)$ and $H_0^1(\mathbb{R}_+)$, and $(\cdot, \cdot)_{L^2}$ denotes the usual $L^2(\mathbb{R}_+)$-inner product. We also note that $B$ is identified as a usual expression $B\psi = -\nu\psi_{xx}$ if $\psi\in \widetilde{H}^2(\mathbb{R}_+)$ holds. Then, we define weak solutions as follows.

Definition 2.3. Let $T > 0$ and let $\tilde{\psi} = \tilde{\psi}(t, x)$ be a given function. For given $\phi_0\in L^2(\mathbb{R}_+)$ and $f\in L^2(0, T; L^2(\mathbb{R}_+))$, we call $\phi$ a weak solution of the initial value problem

if $\phi$ belongs to $X^0(T)$ and satisfies the weak form

for all $\varphi\in C^1_0([0, T)\times \mathbb{R_+})$.

Definition 2.4. Let $T > 0$ and let $\tilde{\phi} = \tilde{\phi}(t, x)$ be a given function. For given $\psi_0\in L^2(\mathbb{R}_+)$ and $g\in L^2(0, T; H^{-1}(\mathbb{R}_+))$, we call $\psi$ a weak solution of the initial-boundary value problem

if $\psi$ belongs to $Y^0(T)$ and satisfies the weak form

for all $\varphi\in H_0^1 (\mathbb{R_+})$ and $h\in C^1_0([0, T))$.

The existence, regularity and estimates of weak solutions to (2.1) and (2.3) are stated as the following lemma.

Lemma 2.5. Let $T > 0$ be a positive constant. Assume that $\tilde{\psi} = \tilde{\psi}(t, x)$ satisfies $\tilde{\psi}\in Y^{1}(T)$. Then, for any $\phi_0\in H^k(\mathbb{R}_+)$ and $f\in L^2(0, T; H^k(\mathbb{R}_+))$ with $k = 0, 1$, there exists a unique weak solution $\phi\in X^k(T)$ to satisfying

for all $0\leq t_1\leq t_2\leq T$.

In addition, in the case $k = 1$, $\phi$ belongs to $C^1([0, T]; L^2(\mathbb{R}_+))$ and is controlled as

Here $\tilde{y} = \tilde{y}(\tau; t, x)\in \mathbb{R}_+$ is a unique solution of the problem

Lemma 2.6. Let $T$, $M$ and $m$ be positive constants. Assume that $\tilde{\phi} = \tilde{\phi}(t, x)$ satisfies

Then, for any $\psi_0\in \widetilde{H}^{k}(\mathbb{R}_+)$ and $g\in L^2(0, T; H^{k-1}(\mathbb{R}_+))$ with $k = 0, 1$, there exists a unique weak solution $\psi\in Y^k(T)$ to (2.3) satisfying

for all $0\leq t\leq T$. Here $\mathcal{C}_1(\delta, M, m)$ is a positive constant decreasing in $\delta$, $M$ and increasing in $m$, and $\mathcal{C}_2(\delta, M, m)$ is a positive constant increasing in $\delta$, $M$ and decreasing in $m$.

Lemmas 2.5 and 2.6 are proved by the same method as ^[11], so we omit the details.

Remark 2.7 (ⅰ) If the same assumptions as in Lemma 2.5 with $k = 1$ are satisfied, then (2.2) for any $\varphi\in C^1_0([0, T)\times \mathbb{R_+})$ becomes equivalent to the first equation of (2.1) in $C([0, T]; L^2(\mathbb{R}_+))$.

(ⅱ) If the same assumptions as in Lemma 2.6 with $k = 1$ hold, then (2.4) for any $\varphi\in H_0^1(\mathbb{R}_+)$ and $h\in C^1_0([0, T))$ becomes equivalent to the first equation of (2.3) in $L^2(0, T; L^2(\mathbb{R}_+))$.

3.   Stationary problem

In this section, we discuss the existence and the convergence rate of the stationary solution $(\tilde{\rho}, \tilde{v}, \tilde{F})$ satisfying the following stationary problems (1.7)–(1.11). To solve this problem, we analyze the properties of the solutions and derive the reduced problem. Integrating (1.7) over $[x, \infty)$ for $x > 0$, we have

Letting $x\to0$ in (3.1), we obtain

Therefore, $v_b < 0$ and (1.10) give the fact that $v_+ < 0$ is necessary for the existence of the stationary solution to the problems (1.7)–(1.11). Furthermore, because of (1.10) and (3.1), we regard as $\tilde{v} < 0$.

On the other hand, (1.9) gives $(\tilde{F}/\tilde{v})_x = 0$. Thus, integrating the resultant inequality over $[x, \infty)$ for $x > 0$ and employing (1.10), we have

This equality means that

for any $x\in\mathbb{R}$. Furthermore, we also get

Namely, in the case $F_+ = 0$, our problem is reduced to the stationary problem of the compressible Navier-Stokes equation, which problem was studied in ^[13,21]. Thus, we mainly consider the case $F_+\neq0$ in this paper. We will discuss the case $F_+ = 0$ at the end of this section.

By integrating (1.8) over $[x, \infty)$ for $x > 0$ and substituing (3.1) and (3.2) into (1.10), we arrive at the following problem:

where

Here, we remark that $I_\beta (v_+) = 0$. Our main purpose of this section is to construct the solution to (3.3)–(3.5). To this end, we analyze the profile of $I_\beta(z)$. It is easy to get $\lim_{z\to-0}I_\beta(z) = \infty$ and

For the derivative, we calculate

and this function leads to

Furthermore, we also obtain

for $z < 0$, which means that $I_\beta(z)$ is a convex function. In the case $M_+ > 1$, (3.9) gives

where

In view of the classification of the sign of $I_\beta^\prime(v_+)$, we use the modified Mach number $M_\beta$ to see that the conditions (3.10)–(3.12) are rewritten as

Remark that $\beta_c\leq |v_+/F_+|$. Especially, in the case $0 < \beta < \beta_c$, there exists $v_*$ such that $v_+ < v_* < 0$ and $I_\beta(v_*) = 0$. Furthermore, we also have $I_\beta(z) > 0$ for $z < v_+$ and $I_\beta(z) < 0$ for $v_+ < z < v_*$. These properties are important to solve the problems (3.3)–(3.5).

Then the existence and property for the solution to (3.3)–(3.5) are described as the following key lemma.

Lemma 3.1. Assume $v_+ < 0$. Then, the following facts hold true.

(ⅰ) (subsonic case) Assume that $M_\beta < 1$ holds. Then, the problems (3.3)–(3.5) with $v_+\neq v_b$ has no solution.

(ⅱ) Assume that $M_\beta \ge 1$ holds. Then, the following assertions hold.

(ⅱ-ⅰ) (supersonic case) Suppose $M_\beta > 1$. Then, there exists a unique solution to (3.3)–(3.5) satisfying the following decay estimates if, and only if, $v_b < v_*$:

Furthermore, the solution $\tilde{v}$ monotonically increases if, and only if, $v_b < v_+$, and monotonically decreases if, and only if, $v_+ < v_b < v_*$.

(ⅱ-ⅱ) (transonic case) Suppose $M_\beta = 1$. Then, there exists a unique solution to (3.3)–(3.5) if, and only if, $v_b < v_+$. Furthermore, the solution $\tilde{v}$ monotonically increases and satisfies the following estimates

Remark 3.2. The generalized Mach number $M_\beta$ gives the following characterization. Let $\Omega_{super}$, $\Gamma_{trans}$ and $\Omega_{sub}$ be the sets that

for $(M_+, \beta) \in \mathbb{R}_+ \times \mathbb{R}_+$. Then, we find that the set $\{(M_+, \beta) \, | \, M_+ > 0, \ \beta > 0\}$ is separated to the three sets $\Omega_{super}$, $\Omega_{sub}$ and $\Gamma_{trans}$. More precisely, these sets are drawn in the $(M_+, \beta)$-plain as follows (Figure 1).

Proof. Proof of Lemma 3.1 (ⅰ) Since (3.9), we have $I_\beta^\prime(v_+) > 0$ if $M_\beta < 1$. Therefore, using (3.3), we immediately conclude that $\tilde{v}(x)$ can not approach to $v_+$ as $x \to \infty$, and there does not exist a solution to (3.3)–(3.5) (see also the two graphs of $I_\beta(z)$ displayed in the below figures).

(ⅱ-ⅰ) In the case $M_\beta > 1$, employing (3.3) and the properties of $I_\beta(z)$ mentioned before, we see that there exists a monotonically increasing solution $\tilde{v}$ to (3.3)–(3.5) if $v_b < v_+$. Similarly, we also find that there exists a monotonically decreasing solution $\tilde{v}$ to (3.3)–(3.5) if $v_+ < v_b < v_*$ (see also the graph of $I_\beta(z)$ displayed in the below figure).

The uniqueness and smoothness of the solutions are derived from the standard argument for ordinary differential equations.

We shall derive the convergence estimate. Taylor's theorem gives

for some $\theta \in (0, 1)$. Since $I_\beta(v_+) = 0$, and $I_\beta^\prime(v_+) < 0$ for $M_\beta > 1$, there exist positive constants $\delta_*$, $c_*$ and $C_*$ such that

for $|\tilde{v} - v_+| \le \delta_*$. Thus, combining (3.3) and (3.19), and solving the resultant problem, we obtain

for $|\tilde{v} - v_+| \le \delta_*$. Now, we had already obtained the existence of the global solution $\tilde{v}$ which approached to $v_+$. This means that there exists $x_* > 0$ such that $|\tilde{v}(x) - v_+| \le \delta_*$ for $x \ge x_*$. This fact and (3.20) yields to (3.16). For the higher derivatives of $\tilde{v}$, we can apply the same argument and omit it in detail.

(ⅱ-ⅱ) In the case $M_\beta = 1$, using the same argument as before, we find that there exists a monotonically increasing solution $\tilde{v}$ to (3.3)–(3.5) if, and only if, $v_b < v_+$ (see also the graph of $I_\beta(z)$ displayed in the below figure).

We consider the convergence estimate. In this case, it is not possible to derive the estimate (3.19), and it should be analyzed more carefully. Using (3.18) with $I_\beta(v_+) = I_\beta^\prime(v_+) = 0$ and the convexity of $I_\beta(z)$, we obtain

where $\tilde{c}_*$ and $\tilde{C}_*$ are positive constants defined by

Then, combining (3.3) and (3.21), and solving the resultant problem, we also get

which means (3.16). For the higher derivatives of $\tilde{v}$, we employ (3.3) and (3.21) again and conclude (3.17). □

Lemma 3.1 immediately gives the following proposition for $(\tilde{\rho}, \tilde{v}, \tilde{F})$.

Proposition 3.3. Assume $v_+ < 0$. Then, the following facts hold true.

(ⅰ) (subsonic case) Assume that $M_\beta < 1$ holds. Then, the problems (1.7)–(1.11) with $v_+\neq v_b$ has no solution.

(ⅱ) Assume that $M_\beta \ge 1$ holds. Then, the following assertions hold.

(ⅱ-ⅰ) (supersonic case)Suppose $M_\beta > 1$. Then, there exists a unique solution to (1.7)–(1.11) if, and only if, $v_b < v_*$, and the following decay estimates are satisfied:

Furthermore, the solution $\tilde{v}$ monotonically increases if, and only if, $v_b < v_+$ and monotonically decreases if, and only if, $v_+ < v_b < v_*$.

(ⅱ-ⅱ) (transonic case) Suppose $M_\beta = 1$. Then, there exists a unique solution to (1.7)–(1.11) if, and only if, $v_b < v_+$. Furthermore, the solution $\tilde{v}$ monotonically increases and satisfies the following estimates:

Remark 3.4. (ⅰ) In the case $F_+ = 0$, the solution $\tilde{F}(x)$ to (1.7)–(1.11) becomes $\tilde{F}(x) \equiv 0$.

(ⅱ) The proof of Lemma 3.1 shows that taking $\beta$ large so that $\beta\geq |v_+/F_+|$ at least, the stationary solution does not exist regardless of the Mach number $M_+$ (see for the proof). This indicates that the strong recoiling effect of elastic force disturbs the stationary outflow of the fluid.

(ⅲ) It follows from (3.22) and (3.23) that $(\tilde{\rho}, \tilde{v}, \tilde{F})$ satisfies

for $k = 1, 2$. These estimates will be repeatedly used in Section 5 and Appendix A.

Proof of Proposition 3.3. The equalities (3.1) and (3.2) tell us that $(\tilde{\rho}, \tilde{F})$ is rewritten by $\tilde{v}$, which is the solution to (3.3)–(3.5). Then, it is easy to confirm that $(\tilde{\rho}, \tilde{v}, \tilde{F})$ is a solution to (1.7)–(1.11). Furthermore, employing (3.15), (3.17), and the fact that

the estimates (3.22) and (3.23) are also obtained. □

4.   Main result

This section is devoted to introducing the main result of this paper.

We first introduce the compatibility condition. We once assume that a smooth solution $(\rho, v, F)(t, x)$ of the problems (1.1)–(1.6) exists and the perturbation $(\rho-\tilde{\rho}, v-\tilde{v}, F-\tilde{F})$ belongs to $Z^1(T)$ with some positive time $T > 0$. Then, since $v-\tilde{v}\in C([0, T]; H_0^1(\mathbb{R}_+))$ holds, $(v-\tilde{v})(0, \cdot)$ belongs to $H_0^1(\mathbb{R}_+)$. Therefore, it is necessary to impose the compatibility condition of $0$-th order:

We state the following theorem related to the asymptotic stability of the stationary solution.

Theorem 4.1. Assume that at least one of the following two cases holds:

${\rm (ND)}$ $M_\beta > 1$ and $v_b < v_*$.

${\rm (D)}$ $M_\beta = 1$, $1 < M_+ < \sqrt{\frac{\rho_+ P^{\prime\prime}(\rho_+)}{2 P^\prime(\rho_+)} + 1}$ and $v_b < v_+$.

Suppose that the initial data $(\rho_0, v_0, F_0)$ and the boundary data $v_b$ satisfy (1.4)–(1.6), compatibility condition (4.1) and $(\rho_0 - \tilde{\rho}, v_0-\tilde{v}) \in H^1(\mathbb{R}_+)$. Furthermore, assume that the initial data also satisfies (1.12) in $H^1(\mathbb{R}_+)$. Then, there exists a positive small number $\varepsilon_1 > 0$ such that if

then the problems (1.1)–(1.6) has a unique solution $(\rho, v, F)(t, x)$ satisfying $(\rho-\tilde{\rho}, v-\tilde{v}, F-\tilde{F}) \in C([0, \infty); H^1)$. Furthermore, the solution converges to the stationary solution as a time goes to infinity, that is,

Remark 4.2. (ⅰ) The restriction $1 < M_+ < (\rho_+ (P^{\prime\prime}(\rho_+)/2 P^\prime(\rho_+)) + 1)^{1/2}$ in the condition (D) arises from the convexity of the function $P(\rho)-\beta_c^2\rho_+^2 F_+^2/ \rho$ around $\rho = \rho_+$ to show the stability of the degenerate stationary solution.

(ⅱ) Under the second condition of (1.4), we can reformulate (1.12) as $F_0-\tilde{F} = -(\rho_0-\tilde{\rho})\tilde{F}/\rho_0$, which gives the smallness of $F_0-\tilde{F}$ in $H^1(\mathbb{R}_+)$, provided that $\rho_0-\tilde{\rho}$ is small in $H^1(\mathbb{R}_+)$. This fact and Theorem 4.1 mean the smallness of $(\rho_0 - \tilde{\rho}, v_0-\tilde{v}, F_0-\tilde{F})$ and the asymptotic stability of $(\tilde{\rho}, \tilde{v}, \tilde{F})$. $

5.   Asymptotic stability of non-degenerate stationary solution

Our purpose of this section is to prove Theorem 4.1. The argument is based on the combination of the local existence theory and the corresponding a priori estimate for the solution.

To consider the stability of the stationary solutions, we introduce a new function by

Then, the Eqs (1.1) and (1.3) lead to

For the Eq (5.1), we assign the initial data $J(0, x) = J_0(x)$ in $x \in \mathbb{R}_+$, where $J_0 : = \rho_0 F_0$. The Eq (5.1) has useful properties, and we often utilize (5.1) instead of (1.3). Indeed, the stationary solution $\widetilde{J}(x)$ of (5.1) satisfies $\tilde{v} \widetilde{J}_x = 0$ and this gives $\widetilde{J}(x) = J_+$, where $J_+ : = \rho_+ F_+$. This fact will be used later.

On the other hand, it is useful to employ the different expression for (1.2), that is,

where

Remark that $P^\prime_\beta(\rho) = P^\prime(\rho) + \beta^2 J_+^2/\rho^2 > 0$ holds true.

We set a perturbation from the stationary solution as

Coupling (1.1), (1.2) and (5.1) with (1.4)–(1.11), the perturbation $(\phi, \psi, \zeta)$ satisfies the system

with the initial conditions and boundary conditions at $x = \infty$ and $x = 0$:

Here, the initial perturbation is defined by

and the functions $f_1$ and $f_2$ are given by

Firstly, we mention the local-in-time existence of the solution of (5.2) with boundary conditions (5.4) and (5.5), and initial and compatibility conditions at $t = \tau$:

Here, $\tau\ge0$ is an arbitrary non-negative number.

Proposition 5.1. There exists a positive small number $\varepsilon\in (0, 1)$ independent of $\tau$ such that the following assertion holds true:

Let $\tau\ge0$ and $M_0\in (0, \varepsilon]$. If $(\phi_\tau, \psi_\tau, \zeta_\tau)$ satisfies the compatibility condition of $0$-th order (5.7) and

then there exists $T = T(\delta, M_0) > 0$ independent of $\tau$ such that the problem (5.2) with (5.4)–(5.6) has a unique solution $(\phi, \psi, \zeta)\in Z^1(\tau, \tau+T)$ satisfying

Remark 5.2. Under (5.9), the first and third equations of (5.2) hold in $C([\tau, \tau+T]; L^2(\mathbb{R}_+))$, and the second equation of (5.2) makes sense in $L^2(\tau, \tau+T; L^2(\mathbb{R}_+))$.

Proposition 5.1 will be proved in Appendix A for the case $\tau = 0$ by using the iteration argument.

We next focus on the a priori estimate in the Sobolev space. To state this, we introduce the following notations:

Then, the a priori estimate is summarized by the following proposition.

Proposition 5.3. Suppose that the same assumptions in Theorem 4.1 hold true. Let $(\phi, \psi, \zeta)$ be a solution to (5.2)–(5.5) in a time interval $[0, T]$ with $(\phi, \psi, \zeta)\in Z^1(T)$. Then there exist positive constants $\varepsilon_0$ and $C_0$, such that if $N(T)+\delta \leq \varepsilon_0$, then the following estimates hold uniformly for $t \in [0, T]$:

To construct the a priori estimate, we have to employ the properties for the stationary solutions constructed in Lemma 3.1. Specifically, the following lemma is important to derive the a priori estimate.

Lemma 5.4 (^[13]). Suppose that the same assumptions as in Proposition 5.3 hold. Then, the following estimates are obtained.

(ⅰ) Let $M_\beta > 1$. Then, the following estimates hold true:

for $t \in [0, T]$ and $k, j \in \mathbb{N}$.

(ⅱ) Let $M_\beta = 1$. Then the following estimates hold true:

for $t \in [0, T]$ and $k, j \in \mathbb{N}$ with $k+j \ge 3$.

Proof. The proof is based on (3.22) and (3.23). The argument is the same as the one in ^[13], and we omit it in detail. □

Proposition 3.3 and the smallness assumptions on $\varepsilon_0$ in Proposition 5.3 ensure that if $\varepsilon_0$ is sufficiently small, then (5.11) is obtained, and there exist certain positive constants $c_\rho$, $C_\rho$, $c_v$ and $C_v$ such that

for $(t, x)\in [0, T]\times\mathbb{R}_+$.

Here, we have noticed the facts $\rho_+ -C(\delta+\|\phi\|_{L^\infty(\mathbb{R}_+)})\le \rho\le \rho_+ +C(\delta+\|\phi\|_{L^\infty(\mathbb{R}_+)})$ and $|v_+|-C(\delta+\|\psi\|_{L^\infty(\mathbb{R}_+)})\le -v\le |v_+| +C(\delta+\|\psi\|_{L^\infty(\mathbb{R}_+)})$ by using Proposition 3.3, and then applying Lemma 2.1. To derive (5.10) in Proposition 5.3, we often utilize this boundedness for $\rho$ and $\tilde{\rho}$.

To construct the a priori estimate, we need an important property of $\zeta$. More precisely, the following key lemma is shown.

Lemma 5.5. Suppose that the same assumptions as in Proposition 5.3 hold true. Then, $\zeta(t) \equiv 0$ in $H^1(\mathbb{R}_+)$ for all $t \in [0, T]$.

Proof. In view of Remark 5.2, we first show that $\zeta$ is the weak solution of

In fact, multiplying third equation of (5.2) by an arbitrary function $\varphi\in C_0^1([0, T)\times\mathbb{R}_+)$, integrating over $(0, T)\times\mathbb{R}_+$ and applying integration by parts, we have the following weak form

This gives the desired fact. We then apply Lemma 2.5 with $k = 1, \; \tilde{\psi} = \psi, \; \phi = \zeta, \; t_1 = 0, \; t_2 = t$ and $f = 0$ to obtain

and the Gronwall inequality gives

for $t \in [0, T]$. Therefore, using the condition (1.12), Remark 3.4 (ⅲ) and (5.9), we obtain $\|\zeta(t)\|_{H^1} = 0$ for all $t \in [0, T]$, and this completes the proof. □

Remark 5.6. Lemma 5.5 means that the solution $F$ is represented by $F = \rho_+ F_+/\rho$, once the solution $\rho$ is constructed.

In view of Lemma 5.5, it is enough to concentrate the derivation of a priori estimate for $(\phi, \psi)(t)$ only. We first show the basic estimate for $(\phi, \psi)(t)$. The proof is based on its suitable energy form.

Lemma 5.7. Suppose that the same assumptions as in Proposition 5.3 hold true. Then the following estimate holds:

for $t \in [0, T]$.

Proof. To derive the desired estimate, we employ the energy form. We introduce the useful energy function as follows:

Then the function $\Phi(\rho, \tilde{\rho})$ has the following expansion.

where $w = 1/\rho$ and $\tilde{w} = 1/\tilde{\rho}$. This expression with $\partial_w \Phi(\rho) = - P_\beta(\rho)$ and $\partial_w^2 \Phi(\rho) = \rho^2 P_\beta^\prime(\rho) > 0$ lead to the fact that $\rho \Phi(\rho, \tilde{\rho})$ is equivalent to $|\rho - \tilde{\rho}|^2$ for small $|\rho - \tilde{\rho}|$, and there exist positive constants $c_0$ and $C_0$ such that

This energy function satisfies the following energy form:

where

Integrating (5.18) over $(0, t)\times \mathbb{R}_+$ and employing the boundary conditions (5.4) and (5.5), we get

Because of $v_b < 0$, the second term on the left-hand side of (5.19) is handled as

For the remainder terms $\mathcal{R}_1$ and $\mathcal{R}_2$, we estimate

Therefore, for the non-degenerate case $M_\beta > 1$, we apply (5.12) and obtain

On the other hand, we can not employ the same argument for $\mathcal{R}_2$ in the degenerate case $M_\beta = M_{\beta_c} = 1$. To overcome difficulty, we reformulate $\mathcal{R}_2$ as

Due to $\tilde{v}_x > 0$, the first term of the right hand side in (5.21) is negative. Furthermore, using the fact that $P_{\beta_c}^{\prime\prime}(\rho_+) > 0$ is satisfied if, and only if, $0 < {\beta_c} < \beta_*$, where

the second term of the right hand side in (5.21) is also negative if $0 < {\beta_c} < \beta_*$. Namely, $\mathcal{R}_2$ is estimated as

for suitably small $\delta$ and $|\phi|$. Therefore, using (5.22), (5.20) also holds for the case $M_\beta = M_{\beta_c} = 1$.

For the last term in (5.19), using the fact that $\| Q(\rho, J)\|_{L^2} \le (|J_+| + \|J\|_{L^\infty}) \|\zeta\|_{L^2}/c_\rho$, we obtain

Finally, substituting the above estimate into (5.19), we arrive at the desired estimate and complete the proof. □

The next goal is to derive the estimate for the first-order derivatives of the solution. To obtain the estimate for $\phi_x$, we need to deal with $\phi_{xx}$ and $\phi(t, 0)$ after differentiating the first equation of (5.2) in $x$ and applying integration by parts, formally. However, since we only treat $\phi(t)\in H^1(\mathbb{R}_+)$ for $0\le t\le T$, $\phi_{xx}$ and $\phi(t, 0)$ do not always exist. Therefore, the formal argument cannot make sense in our setting. To overcome this difficulty, we recall Definition 2.3 and Lemma 2.5 to follow the theory of weak solutions to transport equations. This merit is that $\phi_{xx}$ and $\phi(t, 0)$ are not needed in their statements and the proof of the estimate for $\phi_x$. This argument is inspired in ^[12].

Lemma 5.8. Suppose that the same assumptions as in Proposition 5.3 hold true. Then the following estimate holds:

for $t\in[0, T]$.

Proof. We first claim that $\overline{\phi}: = \phi_x \in C([0, T]; L^2(\mathbb{R}_+))$ is the weak solution of

where $\tilde{f}_1: = -v_x\phi_x+(f_1-\rho\psi_x)_x\in L^2(0, T; L^2(\mathbb{R}_+))$. To show this fact, we notice that the first equation of (5.2) holds for almost every $(t, x)\in(0, T)\times\mathbb{R}_+$ because of Remark 5.2. Let $\varphi$ be an arbitrary function belonging to $C_0^1([0, T)\times\mathbb{R}_+)$. Multiplying the first equation of (5.2) by $-\varphi_x$ and integrating over $(0, T)\times\mathbb{R}_+$ we have

Applying integration by parts and using $a(\psi)\varphi_x = (a(\psi)\varphi)_x-v_x\varphi$ in we arrive at the weak form

which yields the desired fact. Therefore, we are able to apply Lemma 2.5 with $k = 0, \; \tilde{\psi} = \psi, \; \phi = \phi_x, \; t_1 = 0, \; t_2 = tt$ and $f = \tilde{f}_1$ to give

for $0\leq t\leq T$. We next eliminate $\rho \phi_x \psi_{xx}$ in the right-hand side of (5.25). Multiplying the second equation in (5.2) by $\rho \phi_x$ yields

Then, integrating above equation over $(0, t)\times\mathbb{R}_+$ and using the formula

we have

Here, the integral of $\rho^2\phi_x\psi_t$ is calculated via mollification with respect to $t$ and integration by parts. This calculation is standard, so we omit the detail. Then, combining (5.25) and (5.26) to eliminate $\rho \phi_x \psi_{xx}$, we get

where

We estimate the remainder term $R_1$. Since $Q(\rho, J)_x = 2J J_x/\rho - (J^2 - J_+^2)\rho_x/\rho^2$, we have

Thus, this estimate and

give

and

Here, Lemmas 3.1 and 5.4, and (5.14) are also applied. Using the first equation in (5.2), we have

and this gives

for all $\sigma > 0$. Thus, the estimates (5.28), –(5.30) and Lemma 5.4 yield

Here, we used the fact that

Therefore, (5.27) with (5.31) leads to

where $\tilde{c}_\rho : = \min_{\rho \in [c_\rho, C_\rho]} \rho P^\prime_\beta(\rho)$. Thus, letting $\sigma$ suitably small, we arrive at the desired estimate (5.23) and the proof is completed. □

We next focus on $\psi_x$ whose estimate is given in the following lemma.

Lemma 5.9. Suppose that the same assumptions as in Proposition 5.3 hold true. Then, the following inequality holds:

for $t\in[0, T]$.

Proof. Multiplying the second equation in (5.2) by $- \psi_{xx}/\rho$ yields

where

Then, integrating this equality over $(0, t) \times \mathbb{R}_+$ and using the formula

we obtain

Here, the integral of $\psi_x\psi_{xx}$ is calculated via mollification with respect to $t$ and integration by parts. This calculation is standard, so we omit the detail. We then estimate the remainder term as $|R_2| \le C (|\phi_{x}| + |\psi_x| + |f_2|) |\psi_{xx}|$, and this gives

for all $\sigma > 0$. The estimate (5.33) with (5.34) leads to the desired estimate (5.32). This completes the proof. □

Proof of Proposition 5.3. Combining (5.15) and (5.23), we have

Furthermore, substituting the resultant estimate and (5.32), we obtain

Namely this gives

for $t \in [0, T]$. Consequently, by using Lemma 5.5 and taking $\varepsilon_0$ suitably small, we arrive at the desired estimate (5.10). This completes the proof. □

In order to complete the proof of Theorem 4.1, it remains to derive the estimate for $F-\tilde{F}$, which is summarized as the following lemma.

Lemma 5.10. Suppose that the same assumptions as in Theorem 4.1 hold. Then, $F$ satisfies

for all $t\in[0, T]$.

Proof. Using (3.2) and Lemma 5.5, $F-\tilde{F}$ is rewritten as

Therefore, we easily see from Proposition 3.3, Proposition 5.3, (5.14) and (5.36) that

for all $t\in[0, T]$. We next study $(F-\tilde{F})_x$. Differentiating (5.36) in $x$ gives

It then follows from Proposition 3.3, Proposition 5.3, (5.14) and (5.36) that

for all $t\in[0, T]$. Consequently, the desired estimate (5.35) immediately follows from (5.37) and (5.38). This completes the proof. □

Proof of Theorem 4.1. To show the global existence result, we use a bootstrap method based on the combination of Proposition 5.1, Proposition 5.3 and Lemma 5.10. Let $\varepsilon_0$ in Proposition 5.3 be $\varepsilon_0\in (0, \varepsilon]$, where $\varepsilon$ is introduced in Proposition 5.1. We then set

We start from the assumptions

Since $\|(\phi_0, \psi_0, \zeta_0)\|_{H^1}\leq \varepsilon_0/6\leq \varepsilon$ clearly holds, we are able to apply Proposition 5.1 with $\tau = 0$, $M_0 = \varepsilon_0/6\; (\leq\varepsilon)$ and $(\phi_\tau, \psi_\tau, \zeta_\tau) = (\phi_0, \psi_0, \zeta_0)$ to show that (5.2)–(5.5) has a unique solution $(\phi, \psi, \zeta)\in Z^1(0, T) = Z^1(T)$ satisfying (5.8) and (5.9) with $T = T(\delta, \varepsilon_0/6)$, $\tau = 0$, $M_0 = \varepsilon_0/6$ and $(\phi_\tau, \psi_\tau, \zeta_\tau) = (\phi_0, \psi_0, \zeta_0)$. Moreover, it follows from (5.9) and $\delta\le \varepsilon_1\le \varepsilon_0/6$ that

Therefore, we can apply Proposition 5.3 to obtain

which means

Here, we use the fact $\varepsilon_1\leq \varepsilon_0/(6C_0)$. Therefore, we are able to construct a unique solution $(\phi^1, \psi^1, \zeta^1)\in Z^1(T, 2T)$ of (5.2)–(5.5) satisfying (5.8) and (5.9) with $\tau = T$, $M_0 = \varepsilon_0/6$ and $(\phi_\tau, \psi_\tau, \zeta_\tau) = (\phi, \psi, \zeta)(T)$ by applying Proposition 5.1 with $\tau = T = T(\delta, \varepsilon_0/6)$, $M_0 = \varepsilon_0/6$ and $(\phi_\tau, \psi_\tau, \zeta_\tau) = (\phi, \psi, \zeta)(T)$. Moreover, we see from (5.9) and $\delta\le \varepsilon_1\le \varepsilon_0/6$ that

Therefore, we are able to extend the solution of (5.2)–(5.5) in $Z^1(T)$ to that in $Z^1(2T)$ by defining $(\phi, \psi, \zeta)(t, x): = (\phi^1, \psi^1, \zeta^1)(t, x)$ for $(t, x)\in[T, 2T]\times\mathbb{R}_+$. Combining (5.39) and (5.40) to confirm

we can apply Proposition 5.3 with $T = 2T$ and notice the fact $\varepsilon_1\leq \varepsilon_0/(6C_0)$ to obtain

which yields

Therefore, it follows from Proposition 5.1 with $\tau = 2T = 2T(\delta, \varepsilon_0/6)$ and $(\phi_\tau, \psi_\tau, \zeta_\tau) = (\phi, \psi, \zeta)(2T)$ that we are able to extend the solution $(\phi, \psi, \zeta)$ of (5.2)–(5.5) in $Z^1(2T)$ to that in $Z^1(3T)$. Consequently, repeating the above argument we have a global-in-time solution $(\phi, \psi, \zeta)$ of (5.2)–(5.5) in $[0, \infty)\times\mathbb{R}_+$ with the desired properties

We next prove the limit (4.2). We see from (5.41) and the definition of $\varepsilon_1$ that $\|(\phi, \psi, \zeta)(t)\|_{H^1}+\delta\leq \varepsilon_0$ holds for all $t\ge 0$. Therefore, Proposition 5.3 leads to

which implies that there exists a monotone increasing sequence $\{t_m\}_{m = 1}^\infty$ with $t_m\to\infty\; (m\to\infty)$ such that the following limit holds:

In view of Lemma 2.1, it suffices to prove $\|(\phi, \psi)_x(t)\|_{L^2}\to0$ as $t\to\infty$. Since $\psi$ belongs to same function space as Lemma 2.2 with an arbitrary $T$, we have the estimate

for $t\geq t_m$. It is directly seen from the second equation of (5.2) and inequalities (5.14), (5.30) and (5.42) that $\psi_t$ satisfies

Therefore, for any $\eta > 0$, there exists $m_1\in\mathbb{N}$ such that if $t\ge t_{m_1}$ holds, then (5.44) leads to $\|\psi_x(t)\|_{L^2}^2 < \eta$, which yields $\|\psi_x(t)\|_{L^2}\to 0$ as $t\to\infty$. To show $\|\phi_x(t)\|_{L^2}\to0$ as $t\to\infty$, we make use the fact that $\phi_x$ is the weak solution of the problem (5.24) for an arbitrary number $T > 0$. Applying Lemma 2.5 with $k = 0, \; \tilde{\psi} = \psi, \; \phi = \phi_x, \; t_1 = t_m, \; t_2 = t$ and $f = \tilde{f}_1$ we obtain

for $t\geq t_m$. It follows from (3.22), (5.14), (5.28), (5.41) and (5.42) that the integrand in the right-hand side of (5.45) satisfies

Therefore, for any $\eta > 0$, there exists $m_2\in\mathbb{N}$ such that if $t\ge t_{m_2}$ holds, then (5.45) gives $\|\phi_x(t)\|_{L^2}^2 < \eta$, which leads to $\|\phi_x(t)\|_{L^2}\to 0$ as $t\to\infty$. Consequently we have $\|(\phi, \psi)(t)\|_{L^\infty}\to0$ as $t\to\infty$. The limit $\|(F-\tilde{F})(t)\|_{L^\infty}\to 0$ as $t\to\infty$ and property $F-\tilde{F}\in C([0, \infty); H^1(\mathbb{R}_+))$ directly follow from (5.35) and (5.36), and hence we arrive at the desired results $(\rho-\tilde{\rho}, v-\tilde{v}, F-\tilde{F})\in C([0, \infty); H^1(\mathbb{R}_+))$ and (4.2). This completes the proof of Theorem 4.1. □

Author contributions

Yusuke Ishigaki and Yoshihiro Ueda: Conceptualization, Formal analysis, Investigation, Methodology, Validation, Writing-original draft, Writing-review & editing. Both authors have read and approved the final version of the manuscript for publication.

Acknowledgments

Y. I. is supported by JSPS KAKENHI Grant Number JP20H00118 and JP23KJ1408. Y. U. is supported by JSPS KAKENHI Grant Number 21K03327 and 21KK0243. The authors would like to thank Professor Yoshiyuki Kagei for his valuable discussion for the proof of Proposition 5.1.

Conflict of interest

The authors declare no conflict of interests.

A.   Proof of Proposition 5.1

In this appendix, we show Proposition 5.1 by using standard iteration method.

Without loss of geneality, we only consider the case $\tau = 0$. Throughout this appendix, we set $u(t, x): = (\phi, \psi, \zeta)(t, x)$ and $u_0(x): = (\phi_0, \psi_0, \zeta_0)(x)$.

In order to perform iteration method, we rewrite (5.2)–(5.5) in the following form

where the functions $a, \; b$ and the operator $B\psi$ are introduced in Section 2, and $g_1$ and $g_2$ are defined by

Our purpose of this appendix is to show the local-in-time solvability of (A.1)–(A.3). We note that the first and third equations of (A.1) are transport equations of $\phi$ and $\zeta$, and the second equation of (A.1) is a parabolic equation of $\psi$. In view of this consideration, we construct the sequence of functions $\{u^{(n)}\}_{n = 0}^\infty \subset Z^1(T)$ with some positive $T$ to approximate $u$ by the iteration argument.

We first define $u^{(0)} = (\phi^{(0)}, \psi^{(0)}, \zeta^{(0)})$ by the following two steps. The first step is to take $\psi^{(0)}$ by solving the problem

The second step is to determine $\phi^{(0)}$ and $\zeta^{(0)}$ by solving the problem

by using $u_0$ and $\psi^{(0)}$. We next set $u^{(n)} = (\phi^{(n)}, \psi^{(n)}, \zeta^{(n)})$, for $n\in\mathbb{N},$ inductively by the following process. Assuming that $u^{(n-1)}$ is obtained, we define $u^{(n)}$ as the solution of the problem

with (A.2) and (A.3). If the obtained sequence $\{u^{(n)}\}_{n = 0}^\infty$ has a limit $u = (\phi, \psi, \zeta)$ in some function space, then it will be a solution to (A.1)–(A.3). To derive this fact, we prepare the following lemma.

Lemma A.1. Let $u_0 = (\phi_0, \psi_0, \zeta_0)$ satisfy the assumptions as in Proposition 5.1 with $\tau = 0$ except the smallness condition $M_0\le \varepsilon$. Then the following assertions hold true.

(ⅰ) The pair of three functions $u^{(0)} = (\phi^{(0)}, \psi^{(0)}, \zeta^{(0)})$ solving (A.4) and (A.5) exists uniquely in $Z^{1}(T_0)$ with some $T_0 = T_0(\delta, M_0)$. Furthermore, $u^{(0)}$ satisfies $u^{(0)} \in Z_{C_0M_0}^1(T_0)\cap X_{3M_0}^1(T_0)^3$, $(\phi^{(0)}, \zeta^{(0)})\in C^1([0, T_0];L^2(\mathbb{R}_+)),$ and

Here, $C_0 = C_0(\delta, M_0)$ is a certain positive constant increasing in $\delta, M_0 > 0$.

(ⅱ) Let $n\in\{0\}\cup\mathbb{N}$. Then, there exists $T_1 = T_1(\delta, M_0)\in (0, T_0]$ independent of $n$ such that (A.6) has a unique solution $u^{(n)} = (\phi^{(n)}, \psi^{(n)}, \zeta^{(n)})$ in $Z^{1}(T_1)$ satisfying $u^{(n)} \in Z_{C_0M_0}^1(T_1)\cap X_{3M_0}^1(T_1)^3$, $(\phi^{(n)}, \zeta^{(n)})\in C^1([0, T_1];L^2(\mathbb{R}_+))$,

Here $T_0$ and $C_0$ are same constants as in (ⅰ).

(ⅲ) Let $n, m\in\mathbb{N}$. Then, there exists $T_2 = T_2(\delta, M_0)\in (0, T_1]$ independent of $n$ and $m$ such that the following estimates hold:

Here, $T_1$ is the same constant as in (ⅱ) , and $C = C(T_2, \delta, M_0)$ is a certain positive constant increasing in $T_2$, $\delta$ and $M_0$.

Before proving Lemma A.1, we prepare the following lemmata which will play a crutial role.

Lemma A.2. Let $T$, $M$ and $m$ be positive constants, and let $\psi_0\in H_0^1(\mathbb{R}_+)$. Then, the following facts are obtained.

(ⅰ) Let $\overline{u} = (\overline{\phi}, \overline{\psi}, \overline{\zeta})$ be a given pair of functions satisfying

for $0\le t\le T$. Then, there exists a unique solution $\psi\in Y^1(T)$ of the problem

Furthermore, $\psi$ satisfies

Here, $\mathcal{C}_1(\delta, M, m)$ is a positive constant taken from (2.7) in Lemma 2.6, and $C_1(\delta, M, m)$ is a positive constant increasing in $\delta, M$ and decreasing in $m$.

(ⅱ) For $j = 1, 2$, let $\overline{u}^{(j)} = (\overline{\phi}^{(j)}, \overline{\psi}^{(j)}, \overline{\zeta}^{(j)})$ be given pairs of functions satisfying (A.9), and let $\psi^{(j)}\in Y^1(T)$ be solutions of (A.10) with $\overline{u} = \overline{u}^{(j)}$ respectively. Then, $\psi^{(1)}-\psi^{(2)}$ satisfies

Here, $C_2(T, \delta, M, m)$ is a positive constant increasing in $T, \; \delta, \; M$ and decreasing in $m$.

Lemma A.3. Let $T$, $\widetilde{M}$ and $M$ be positive constants, and let $\phi_0, \zeta_0\in H^1(\mathbb{R}_+)$. Then, the following facts are obtained.

(ⅰ) Let $\overline{u} = (\overline{\phi}, \overline{\psi}, \overline{\zeta})$ be a given pair of functions satisfying $\overline{u}\in Z_{\widetilde{M}}^1(T)\cap X_M^1(T)^3$. Then, there exists a unique pair of solutions $(\phi, \zeta)\in X^1(T)^2$ of the two problems

and

Furthermore, $\phi$ and $\zeta$ satisfy $\phi, \zeta\in C^1([0, T]; L^2(\mathbb{R}_+))$ and the following estimates

Here, $C_3(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M$.

(ⅱ) For $j = 1, 2$, let $\overline{u}^{(j)} = (\overline{\phi}^{(j)}, \overline{\psi}^{(j)}, \overline{\zeta}^{(j)})$ be given pairs of functions satisfying $\overline{u}^{(j)}\in Z_{\widetilde{M}}^1(T)\cap X_M^1(T)^3$, and let $(\phi^{(j)}, \zeta^{(j)})\in (X^1(T))^2$ be pairs of solutions of (A.13) and (A.14) with $\overline{u} = \overline{u}^{(j)}$ respectively.

Then, $\phi^{(1)}-\phi^{(2)}$ and $\zeta^{(1)}-\zeta^{(2)}$ satisfy

Here, $C_4(T, \delta, \widetilde{M}, M)$ and $C_5(\delta, M)$ are positive constants increasing in $T, \; \delta, \; \widetilde{M}, \; M$.

Proof of Lemma A.2. (ⅰ) We first note that $g_2(\overline{u}, \overline{u}_x)$ belongs to $L^2(0, T; L^2(\mathbb{R}_+))$ by using (A.9). Then, applying Lemma 2.6 with $\tilde{\phi} = \overline{\phi}$, $g = g_2(\overline{u}, \overline{u}_x)$ and $k = 1$, the problem (A.10) has a unique solution $\psi\in Y^1(T)$ satisfying

for $0\leq t\leq T$. Here $\mathcal{C}_1(\delta, M, m)$ and $\mathcal{C}_2(\delta, M, m)$ are taken in (2.7). We then use the Gronwall inequality to earn

Therefore, since $g_2(\overline{u}, \overline{u}_x)$ satisfies

with some constant $C_{1, 1}(\delta, M, m)$ increasing in $\delta, M$ and decreasing in $m$, we arrive at the desired inequality (A.11). This completes the proof of (ⅰ).

(ⅱ) We first see from (A.10) that $\psi^{(1)}-\psi^{(2)}$ is a weak solution of

Applying (2.7) in Lemma 2.6 with $k = 0$, $\tilde{\phi} = \overline{\phi}^{(1)}$, $\psi = \psi^{(1)}-\psi^{(2)}$, $g = g_2(\overline{u}^{(1)}, \overline{u}_{x}^{(1)})-g_2(\overline{u}^{(2)}, \overline{u}_{x}^{(2)})-(\overline{\phi}^{(1)}-\overline{\phi}^{(2)})\psi_t^{(2)}$, we obtain

for $0\leq t\leq T$, where $C_{2, 1}(\delta, M, m)$ is a positive constant increasing in $\delta, M$ and decreasing in $m$. We then use the Gronwall inequality to rewrite this inequality as

We next estimate the right-hand side of (A.19). The function $(\bar{\phi}^{(1)}-\bar{\phi}^{(2)})\psi_t^{(2)}$ is controlled as

Using (A.9) with $\overline{u} = \overline{u}^{(j)}$ with $j = 1, 2$ and integration by parts, $g_2(\overline{u}^{(1)}, \overline{u}_x^{(1)})-g_2(\overline{u}^{(2)}, \overline{u}_x^{(2)})$ is estimated as

where $C_{2, 2}(\delta, M, m)$ is a positive constant increasing in $\delta, M$ and decreasing in $m$. Therefore, together with (A.19) and the estimates for $(\overline{\phi}^{(1)}-\overline{\phi}^{(2)})\psi_t^{(2)}$ and $g_2(\overline{u}^{(1)}, \overline{u}_x^{(1)})-g_2(\overline{u}^{(2)}, \overline{u}_x^{(2)})$, we earn (A.12). This completes the proof of (ⅱ). □

Proof of Lemma A.3. (ⅰ) We first note that $g_1(\overline{\phi}, \overline{\psi}, \overline{\psi}_x)$ belongs to $L^2(0, T; H^1(\mathbb{R}_+))$ and $a(\overline{\psi})$ satisfies the same assumptions for $\tilde{\psi}$ as in Lemma 2.5 by using the assumptions of $\bar{u}$. We then apply Lemma 2.5 with $k = 1$, $\tilde{\psi} = \overline{\psi}$ and $f = g_1(\overline{\phi}, \overline{\psi}, \overline{\psi}_x)$ to show that (A.13) has a unique solution $\phi\in X^1(T)$ with $\phi\in C^1([0, T]; L^2(\mathbb{R}_+))$. As in this argument, (A.14) has a unique solution $\zeta\in X^1(T)$ with $\zeta\in C^1([0, T]; L^2(\mathbb{R}_+))$ by replacing $\phi$ and $g_1(\overline{\phi}, \overline{\psi}, \overline{\psi}_x)$ as $\zeta$ and $0$, respectively.

We next derive (A.15). It follows from Lemma 2.1 and (2.5) in Lemma 2.5 with $k = 1$, $\tilde{\psi} = \overline{\psi}$, $f = g_1(\overline{\phi}, \overline{\psi}, \overline{\psi}_x)$, $t_1 = 0$ and $t_2 = t$ that $\phi$ satisfies

for $0\leq t\leq T$, where $C_{3, 1}(\delta)$ is a positive constant increasing in $\delta$. Applying the Gronwall inequality to the resultant inequality and using the fact $\int_0^t\|\bar{\psi}_x(\tau)\|_{H^1}d\tau\leq \sqrt{T}\|\overline{u}\|_{Z^1(T)}\leq \sqrt{T}\widetilde{M}$ yields

for $0\leq t\leq T$, where $C_{3, 2}(T, \delta, \widetilde{M}): = C_{3, 1}(\delta)\sqrt{T}+\widetilde{M}$. Similarly, $\zeta$ is estimated as

for $0\leq t\leq T$. We then add these inequalities to give

for $0\leq t\leq T$. To complete the derivation of (A.15), it remains to estimate the right-hand side of this inequality. From the assumption of $\overline{u}$, $g_1(\overline{\phi}, \overline{\psi}, \overline{\psi}_x)$ satisfies

where $C_{3, 3}(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M > 0$. Hence (A.20) leads to

which yields (A.15). Here $C_{3, 4}(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M$. We finally show (A.16) via a characteristic method. Let $\overline{y} = \overline{y}(\tau; t, x)\in \mathbb{R}_+$ be a unique solution of

Decomposing $\phi(t, x) = \phi_0(\overline{y}(0;t, x))+(\phi(t, x)-\phi_0(\overline{y}(0;t, x)))$ and using (2.6) in Lemma 2.5 and (A.20) immediately yields

which leads to (A.16). This completes the proof of (ⅰ).

(ⅱ) We first see from (A.13) that $\phi^{(1)}-\phi^{(2)}$ is a weak solution of

Applying (2.5) in Lemma 2.5 with $k = 0$, $\tilde{\psi} = \overline{\psi}$, $\phi = \phi^{(1)}-\phi^{(2)}$, $f = -(\overline{\psi}^{(1)}-\overline{\psi}^{(2)})\phi^{(2)}_{x}+g_1(\overline{\phi}^{(1)}, \overline{\psi}^{(1)}, \overline{\psi}_x^{(1)})-g_1(\overline{\phi}^{(2)}, \overline{\psi}^{(2)}, \overline{\psi}_x^{(2)})$, $t_1 = 0$ and $t_2 = T$, we obtain

for $0\leq t\leq T$, where $C_{4, 1}(\delta)$ is a positive constant depending only on $\delta$. We then use the Gronwall inequality to read the resultant inequality as

for $0\leq t\leq T$. Here $C_{4, 2}(T, \delta, \widetilde{M})$ is a positive constant increasing in $T, \delta, \widetilde{M}$. We next focus on the right-hand side of (A.21). We directly compute the term involving $(\overline{\psi}^{(1)}-\overline{\psi}^{(2)})\phi_x^{(2)}$ as

where $C_{4, 3}(T, \delta, \widetilde{M}, M)$ denotes $C_{4, 3}(T, \delta, \widetilde{M}, M): = 2C_{4, 1}(\delta)e^{2C_{4, 2}(T, \delta, \widetilde{M})\sqrt{T} }$. For the term containing $g_1(\overline{\phi}^{(1)}, \overline{\psi}^{(1)}, \overline{\psi}_x^{(1)})-g_1(\overline{\phi}^{(2)}, \overline{\psi}^{(2)}, \overline{\psi}_x^{(2)})$, we use the assumptions of $\bar{u}^{(j)}\; (j = 1, 2)$ to deduce

where $C_{4, 4}(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M$. This estimate leads to

where $C_{4, 5}(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M$. Therefore, combining (A.21) and these estimates, we arrive at

Here, $C_{4, 6}(T, \delta, \widetilde{M}, M)$ is a positive constant increasing in $T, \delta, \widetilde{M}, M$. Similarly, the following estimate for $\zeta^{(1)}-\zeta^{(2)}$ is obtained by replacing $\phi^{(j)}$ and $g_1(\overline{\phi}^{(j)}, \overline{\psi}^{(j)}, \overline{\psi}_x^{(j)})$ as $\zeta^{(j)}$ and $0$, respectively for $j = 1, 2$ in the above argument:

Hence, combining the above two estimates leads to (A.17). We finally prove (A.18) by using the equality

for any $\varphi\in H_0^1(\mathbb{R}_+)$ with $\|\varphi\|_{H^1}\le 1$. By virtue of $\overline{u}^{(j)}\in X_M^1(T)^3, \; j = 1, 2$ and integration by parts, each terms in the right-hand side of this equality are controlled as

for $0\le t \le T$, where $C_5(\delta, M)$ is a positive constant increasing in $\delta, M$. Therefore, using these estimates, we have (A.18) This completes the proof of (ⅱ). □

Proof of Lemma A.1. (ⅰ) We first focus on the unique existence and properties of $\psi^{(0)}$. Let $T_{0, 1} > 0$ be a constant taken suitably small later. Since $u_0$ satisfies (A.9) in Lemma A.2 with $\bar{u} = u_0$, $T = T_{0, 1}$, $M = M_0$ and $m = 1/2$, (A.4) has a unique solution $\psi^{(0)}\in Y^1(T_{0, 1})$. Furthermore, using (A.11) with $T = T_{0, 1}$, the following estimate for $\psi = \psi^{(0)}$ is obtained:

Therefore using the properties $\mathcal{C}_1(\delta, M_0, 1/2)\ge \mathcal{C}_1(\delta, 3M_0, 1/4)$ and $C_1(\delta, M_0, 1/2)\le C_1(\delta, 3M_0, 1/4)$, and then letting $T_{0, 1} = T_{0, 1}(\delta, M_0)$ small such that $e^{ C_1(\delta, 3M_0, 1/4)T_{0, 1} }(M_0^2+C_1(\delta, 3M_0, 1/4)T_{0, 1}) \leq 9M_0^2$, we have

which yields $\psi^{(0)}\in Y_{\widetilde{M}_0}^1(T_{0, 1})\cap X_{3M_0}^1(T_{0, 1})$. Here $\widetilde{M}_0$ denotes $\widetilde{M}_0: = 3\{1+\mathcal{C}_1(\delta, 3M_0, 1/4)^{-1}\}^{1/2}M_0$. We next investigate the existence and properties of $(\phi^{(0)}, \zeta^{(0)})$. Let $\widetilde{u}^{(0)} = \widetilde{u}^{(0)}(t, x)$ be $\widetilde{u}^{(0)}: = (\phi_0, \psi^{(0)}, \zeta_0)$ and let $T_{0}$ be a constant $T_{0} \in (0, T_{0, 1}]$ determined later. Then in view of (A.22), $\widetilde{u}^{(0)}$ satisfies $\widetilde{u}^{(0)}\in Z_{\widetilde{M}_0}^1(T_{0})\cap X_{3M_0}^1(T_{0})^3$. We then apply Lemma A.3 (ⅰ) with $\overline{u} = \widetilde{u}^{(0)}$, $T = T_{0}$, $\widetilde{M} = \widetilde{M}_0$ and $M = 3M_0$ to show that (A.5) has a unique solution $(\phi^{(0)}, \zeta^{(0)})\in X^1(T_{0})^2$ satisfying $\phi^{(0)}, \zeta^{(0)}\in C^1([0, T_0];L^2(\mathbb{R}_+))$ and

Here, $\widetilde{C}_{3, 0}(T_{0}, \delta, M_0)$ is defined as $\widetilde{C}_{3, 0}(T_{0}, \delta, M_0): = C_3(T_{0}, \delta, \widetilde{M}_0, 3M_0)$ by using $C_3(T, \delta, \widetilde{M}, M)$ defined in Lemma A.3 (ⅱ). We also note that $\widetilde{C}_{3, 0}(T_{0}, \delta, M_0)$ increases $in T_{0}, \delta, M_0$. Therefore taking $T_{0} = T_{0}(\delta, M_0)$ small such that $e^{ \widetilde{C}_{3, 0}(T_{0}, \delta, M_0)\sqrt{T_{0}} }(4M_0^2+\widetilde{C}_{3, 0}(T_{0}, \delta, M_0)\sqrt{T_{0}}) \leq 9M_0^2$ and $\widetilde{C}_{3, 0}(T_{0}, \delta, M_0)\sqrt{T_{0}} \le (1/4)\inf_{x\in\mathbb{R}_+}\tilde{\rho}(x)$ in the above inequalities, we have

As a result, adding (A.22) to (A.23) imply $u^{(0)}\in Z_{C_0 M_0}^1(T_{0})\cap X_{3M_0}^1(T_{0})^3$. Here $C_0 = C_0(\delta, M_0)$ is given by $C_0: = 3\{3+\mathcal{C}_1(\delta, 3M_0, 1/4)^{-1}\}^{1/2}$ increasing in $\delta, M_0$. This completes the proof of Lemma A.1 (ⅰ).

(ⅱ) We employ the induction argument with respect to $n$. The case $n = 0$ is already true. We assume that the properties $u^{(n-1)}\in Z_{C_0 M_0}^1(T_{1})\cap X_{3M_0}^1(T_{1})^3$, $\phi^{(n-1)}, \zeta^{(n-1)}\in C^1([0, T_1];L^2(\mathbb{R}_+))$ and

hold with some $n\in\mathbb{N}$ and $T_1 = T_1(\delta, M_0) \in (0, T_0]$ independent of $n$. Then using Lemma A.2 (ⅰ) and Lemma A.3 (ⅰ), $u^{(n)} = (\phi^{(n)}, \psi^{(n)}, \zeta^{(n)})$ is uniquely determined by solving (A.6) and satisfies $u^{(n)}\in Z^1(T_1)$ and $\phi^{(n)}, \zeta^{(n)}\in C^1([0, T_1];L^2(\mathbb{R}_+))$. In addition, it follows (A.11) and (A.15) with $u = u^{(n)}$, $\overline{u} = u^{(n-1)}$, $T = T_1$, $\widetilde{M} = C_0 M_0$, $M = 3M_0$ and $m = 1/4$ that

Here, $\widetilde{C}_{3}(T_{1}, \delta, M_0)$ is defined as $\widetilde{C}_{3}(T_{1}, \delta, M_0): = C_3(T_{1}, \delta, C_0M_0, 3M_0)$ by using $C_3(T, \delta, \widetilde{M}, M)$ defined in Lemma A.3 (ⅰ). Therefore, taking $T_1 = T_1(\delta, M_0)$ suitably small without the dependence of $n$, $u^{(n)}$ also belongs to $Z_{C_0 M_0}^1(T_1)\cap X_{3M_0}^1(T_1)^3$, and satisfies

As a result, the statement for $n$ becomes true. This completes the proof of Lemma A.1 (ⅱ).

(ⅲ) We first estimate $\psi^{(n+1)}-\psi^{(n)}$. Replacing $(\psi^{(1)}, \psi^{(2)})$ and $(\overline{u}^{(1)}, \overline{u}^{(2)})$ as $(\psi^{(n+1)}, \psi^{(n)})$ and $({u}^{(n)}, {u}^{(n-1)})$ in Lemma A.2 (ⅱ) respectively, (A.12) is read as

Here, we set $\widetilde{C}_2(T_2, \delta, M_0): = C_2(T_2, \delta, 3M_0, 1/4)$. Then, using $u^{(n)}\in Z_{C_0M_0}^1(T_2)$ for all $n\in\mathbb{N}\cup\{0\}$ to earn $\|\psi^{(n)}\|_{Y^1(T)}\leq C_0M_0$, we have

where $C_6(T_2, \delta, M_0)$ is a positive constant increasing in $T_2, \delta, M_0$. We next focus on $\phi^{(n+1)}-\phi^{(n)}$ and $\zeta^{(n+1)}-\zeta^{(n)}$. Changing $(\phi^{(1)}, \phi^{(2)})$, $(\zeta^{(1)}, \zeta^{(2)})$ and $(\bar{u}^{(1)}, \bar{u}^{(2)})$ as $(\phi^{(n+1)}, \phi^{(n)})$, $(\zeta^{(n+1)}, \zeta^{(n)})$ and $(u^{(n)}, u^{(n-1)})$ in (A.17) in Lemma A.3 (ⅱ), we have

Here, we put $\widetilde{C}_4(T_2, \delta, M_0): = C_4(T_2, \delta, C_0M_0, 3M_0)$. Then noticing from (ⅱ) of this lemma that $u^{(n)}\in X_{3M_0}^1(T_2)^3$ holds for all $n\in\mathbb{N}$, this inequality is rewritten as

where $C_7(T_2, \delta, M_0)$ is a positive constant increasing in $T_2, \delta, M_0$. Consequently, combining the above estimates we arrive at (A.7). Similarly (A.8) is obtained by replacing $(\phi^{(1)}, \phi^{(2)})$, $(\bar{u}^{(1)}, \bar{u}^{(2)})$ and $M$ as $(\phi^{(n)}, \phi^{(m)})$, $(u^{(n-1)}, u^{(m-1)})$ and $3M_0$, respectively in (A.17) in Lemma A.3 (ⅱ) to derive

for any $n, m\in\mathbb{N}$, and then applying the fact $u^{(m)}\in X_{3M_0}^1(T_2)^3$ for any $m\in\mathbb{N}$. This completes the proof of Lemma A.1 (ⅲ). □

Proof of Proposition 5.1. Let $\varepsilon$ and $T_2$ be the constants stated in Proposition 5.1 and Lemma A.1 (ⅲ). For simplicity, we rewrite $T_2$ as $T$, and let $\varepsilon$ and $T$ be $\varepsilon, T\in (0, 1)$.

We first see from (A.7) in Lemma A.1 (ⅲ) that if $u_0$ satisfies $\|u_0\|_{H^1}\leq M_0\leq \varepsilon$, then $\{u^{(n)}\}_{n = 0}^\infty$ is a Cauchy sequence in $Z^0(T)$, provided that $\varepsilon$ and $T = T(\delta, M_0)$ are chosen as $C(1, \delta, 1)(\varepsilon^2+T) < 1$, respectively. Furthermore, owing to this fact and (A.8), $\{\phi_t^{(n)}\}_{n = 0}^\infty$ becomes a Cauchy sequence in $C([0, T]; H^{-1}(\mathbb{R}_+))$. Therefore $\{u^{(n)}\}_{n = 0}^\infty$ has a limit $u = (\phi, \psi, \zeta)$ such that

Moreover, Lemma A.1 (ⅱ) and (A.24) lead to

and it follows from Lemma A.1 (ⅱ) that there exists a subsequence $\{u^{(n_k)}\}_{k = 0}^\infty\subset \{u^{(n)}\}_{n = 0}^\infty$ satisfying

We are now going to claim that $u$ is a solution of (A.1) in $(0, T)\times\mathbb{R}_+$ with $u\in Z^1(T)$. We first investigate $\phi$ and $\zeta$. Since the first equation of (A.6) is satisfied in $C[0, T]; L^2(\mathbb{R}_+))$ for $n\in\mathbb{N}$, the following weak form holds for any $\varphi\in C^1_0([0, T)\times \mathbb{R_+})$:

Therefore, it is not difficult to see from taking the limit $n\to\infty$ in this equality and using Lemma A.1 (ⅱ) and (A.24) that $\phi$ satisfies

This means that $\phi$ is a weak solution of (2.1) with $\tilde{\psi} = \psi$ and $f = g_1(\phi, \psi, \psi_x)$ in Definition 2.3.

Similarly, since $\zeta^{(n)}$ solves the third equation of (A.6) for $n\in\mathbb{N}$ in $C[0, T]; L^2(\mathbb{R}_+))$, $\zeta$ satisfies

This implies that $\zeta$ is a weak solution of (2.1) with $\tilde{\psi} = \psi$ and $f = 0$ in Definition 2.3.

We next check $g_1(\phi, \psi, \psi_x) \in L^2(0, T; H^1(\mathbb{R}_+))$ and $\psi\in Y^1(T)$, which imply $\phi, \zeta \in X^1(T)$ and $\phi, \zeta \in C^1([0, T]; L^2(\mathbb{R}_+))$ by applying the regularity properties of weak solutions in Lemma 2.5. From (A.24)–(A.26), $g_1(\phi, \psi, \psi_x) \in L^2(0, T; H^1(\mathbb{R}_+))$ holds true. Moreover $\psi\in Y^1(T)$ is also satisfied by (A.24), (A.26) and $C([0, T]; L^2(\mathbb{R}_+))\cap L^2(0, T; \widetilde{H}^2(\mathbb{R}_+))\subset C([0, T]; H_0^1(\mathbb{R}_+))$. Therefore $\phi, \zeta \in X^1(T)$ and $\phi, \zeta \in C^1([0, T]; L^2(\mathbb{R}_+))$ hold ture, and the properties $\phi, \zeta \in X_{3M_0}^1(T)$ and $\inf_{(t, x)\in [0, T]\times \mathbb{R}_+}\phi (t, x)\ge -(3/4)\inf_{x\in \mathbb{R}_+}\tilde{\rho}(x)$ follow from Lemma A.1 (ⅱ), (A.24) and (A.25). As a result, restricting $\varphi\in C^1_0((0, T)\times \mathbb{R_+})$ in (A.27) and (A.28), and then applying integration by parts, we see that $u = (\phi, \psi, \zeta)$ satisfies the first and third equations of (A.1) in $C([0, T]; L^2(\mathbb{R}_+))$.

We next study $\psi$. Since $\psi^{(n)}$ is the solution of the second equation of (A.6) for $n\in\mathbb{N}$, the following weak form holds for all $\varphi\in H^1_0 (\mathbb{R_+})$ and $h\in C^1_0([0, T))$:

Therefore taking the limit $n\to\infty$ in this equality and using Lemma A.1 (ⅱ) and (A.24), we obtain

This shows that $\psi$ is a weak solution of (2.3) with $\tilde{\phi} = \phi$ and $g = g_2(u, u_x)$ in Definition 2.4. The properties $\psi\in Y_{C_0M_0}^1(T)\cap X_{3M_0}^1(T)$ and $g_2(u, u_x)\in L^2(0, T; L^2(\mathbb{R}_+))$ are confirmed by Lemma A.1 (ⅱ), (A.25) and (A.26). From the above discussion, the assumptions for $\tilde{\phi} = \phi$ in Lemma 2.6 with $M = 3M_0$ and $m = 1/4$ are also justified. Therefore, we gurantee these properties and then obtain the second equation of (A.1) in $L^2(0, T; L^2(\mathbb{R}_+))$ by restricting $h\in C^1_0(0, T)$ in (A.29) and applying integration by parts.

Consequently, we prove the local-in-time existence of the solution to (5.2) with (5.4)–(5.6) satisfying $u\in X_{3M_0}^1(T)$, (5.8) and (5.9) with $\tau = 0$. The uniqueness of the solution is confirmed in a similar manner to the proof of Lemma A.1 (ⅲ). This completes the proof of Proposition 5.1. □