Global well-posedness of non-heat conductive compressible Navier-Stokes equations in 1D

TL;DR

This paper proves the global existence and uniqueness of solutions for the 1D compressible Navier-Stokes equations without heat conduction, allowing initial vacuum and nonnegative density, thus extending previous heat-conductive results.

Contribution

It establishes the global well-posedness for the 1D full compressible Navier-Stokes equations with zero heat conductivity, generalizing prior results to include initial vacuum and non-heat conductive cases.

Findings

Global well-posedness for any $H^1$ initial data.

Initial vacuum is permitted, density need not be uniformly positive.

Extension of Kazhikhov--Shelukhin's result to non-heat conductive case.

Abstract

In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: \emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.\,Appl.\,Math.\,Mech., \bf41 \rm(1977), 273--282.) from the heat conductive case to the non-heat conductive case, and the initial vacuum is allowed.

Equations

$${\rho }_t+(\rho u{)}_x$$

$$\rho (u_t+u{u}_x)-\mu u_{xx}+p_x$$

$$c_v\rho ({\theta }_t+u{\theta }_x)+u_{x}p$$

$$p_t+u{p}_x+\gamma u_{x}p=\mu (\gamma -1)(u_x{)}^2,$$

$${\rho }_t+(\rho u{)}_x$$

$$\rho (u_t+u{u}_x)-\mu u_{xx}+p_x$$

$$p_t+u{p}_x+\gamma u_{x}p$$

$$\left\{\begin{array}[]{l}\eta_{t}(y,t)=u(\eta(y,t),t),\\ \eta(y,0)=y.\end{array}\right.$$

$$\varrho (y,t):=\rho (\eta (y,t),t),v(y,t):=u(\eta (y,t),t),\pi (y,t):=p(\eta (y,t),t),$$

$$J_t=v_y,$$

$${\varrho }_t+\frac{v_y}{J}\varrho$$

$$\varrho v_t-\frac{\mu }{J}{\left(\frac{v_y}{J}\right)}_y+\frac{{\pi }_y}{J}$$

$${\pi }_t+\gamma \frac{v_y}{J}\pi$$

$$(J\varrho {)}_t=J_t\varrho +J{\varrho }_t=v_y\varrho -J\frac{v_y}{J}\varrho =0,$$

$${\varrho }_0{v}_t-\mu {\left(\frac{v_y}{J}\right)}_y+{\pi }_y=0.$$

$$J_t$$

$${\varrho }_0{v}_t-\mu {\left(\frac{v_y}{J}\right)}_y+{\pi }_y$$

$${\pi }_t+\gamma \frac{v_y}{J}\pi$$

$$v(0,t)=v(L,t)=0$$

$$(J,{\varrho }_{0}v,\pi ){|}_{t=0}=(1,{\varrho }_0{v}_0,{\pi }_0).$$

$$0<J\in C([0,T];H^1),$$

$${\varrho }_{0}v\in C([0,T];L^2),$$

$$\sqrt{{\varrho }_0}{v}_t\in L^2(0,T;L^2),$$

$$0\le \pi \in C([0,T];H^1),$$

$$\left(\mu\frac{v_{y}}{J}-\pi\right)\Big{|}_{y=0,L}=0.$$

$$0<\underline{\varrho }\le {\varrho }_0\le \bar{\varrho }<\infty ,0<\underline{J}\le J_0\le \bar{J}<\infty ,$$

$${\pi }_0\ge 0,({\varrho }_0,J_0,{\pi }_0)\in H^1,v_0\in H_0^1,$$

$$0<J\in C([0,T_0];H^1),J_t\in L^{\infty }(0,T_0);L^2),$$

$$v\in C([0,T_0];H_0^1)\cap L^2(0,T_0;H^2),v_t\in L^2(0,T_0;L^2),$$

$$0\le \pi \in C([0,T_0];H^1),{\pi }_t\in L^{\infty }(0,T_0;L^2).$$

$${\int }_0^{L}J(y,t)dy={\ell }_0$$

$$\left({\int }_0^L\left(\frac{{\varrho }_0}{2}{v}^2+\frac{J\pi }{\gamma -1}\right)dy\right)(t)=E_0,$$

$$\frac{1}{2}\frac{d}{dt}{\int }_0^L{\varrho }_0{v}^{2}dy+\mu {\int }_0^L\frac{(v_y{)}^2}{J}dy={\int }_0^L{v}_y\pi dy.$$

$$\frac{d}{dt}{\int }_0^{L}J\pi dy+(\gamma -1){\int }_0^L{v}_y\pi dy=\mu (\gamma -1){\int }_0^L\frac{(v_y{)}^2}{J}dy,$$

$$\frac{d}{dt}{\int }_0^L\left(\frac{{\varrho }_0}{2}{v}^2+\frac{J\pi }{\gamma -1}\right)dy=0,$$

$${\varrho }_0{v}_t-\mu (\log J{)}_{yt}+{\pi }_y=0.$$

$${\varrho }_0(v-v_0)-\mu (\log J-\log J_0{)}_y+{\left({\int }_0^t\pi d\tau \right)}_y=0,$$

Full text

1. Introduction

1.1. The compressible Navier-Stokes equations

The one-dimensional non-heat conductive compressible Navier-Stokes equations read as

[TABLE]

where $\rho,u,\theta$, and $p$, respectively, denote the density, velocity, absolute temperature, and pressure. The viscous coefficient $\mu$ is assumed to be a positive constant. The state equation for the ideal gas reads as $p=R\rho\theta,$ where $R$ is a positive constant. Using the state equation, one can derive from (1.1) and (1.3) that

[TABLE]

where $\gamma-1=\frac{R}{c_{v}}.$ Therefore, we have the follow system

[TABLE]

The compressible Navier-Stokes equations have been extensively studied. In the absence of vacuum, i.e., the case that the density has a uniform positive lower bound, the local well-posedness was proved long time ago by Nash [40], Itaya [18], Vol’pert-Hudjaev [51], Tani [44], Valli [45], and Lukaszewicz [35]; uniqueness was proved even earlier by Graffi [14] and Serrin [43]. Global well-posedness of strong solutions in 1D has been well-known since the works by Kanel [22], Kazhikhov–Shelukhin [24], and Kazhikhov [23]; global existence and uniqueness of weak solutions was also established, see, e.g., Zlotnik–Amosov [52, 53], Chen–Hoff–Trivisa [1], and Jiang–Zlotnik [21], and see Li–Liang [29] for the result on the large time behavior. The corresponding global well-posedness results for the multi-dimensional case were established only for small perturbed initial data around some non-vacuum equilibrium or for spherically symmetric large initial data, see, e.g., Matsumura–Nishida [36, 37, 38, 39], Ponce [41], Valli–Zajaczkowski [46], Deckelnick [9], Jiang [19], Hoff [15], Kobayashi–Shibata [25], Danchin [7], Chen-Miao-Zhang [2], Chikami–Danchin [3], Dachin-Xu [8], Fang-Zhang-Zi [10], and the references therein.

In the presence of vacuum, that is the density may vanish on some set or tends to zero at the far field, global existence of weak solutions to the isentropic compressible Navier-Stokes equations was first proved by Lions [33, 34], with adiabatic constant $\gamma\geq\frac{9}{5}$, and later generalized by Feireisl–Novotný–Petzeltová [11] to $\gamma>\frac{3}{2}$, and further by Jiang–Zhang [20] to $\gamma>1$ for the axisymmetric solutions. For the full compressible Navier-Stokes equations, global existence of the variational weak solutions was proved by Feireisl [12, 13], which however is not applicable for the ideal gases. Local well-posedness of strong solutions to the full compressible Navier-Stokes equations, in the presence of vacuum, was proved by Cho–Kim [6], see also Salvi–Stra$\check{\text{s}}$kraba [42], Cho–Choe–Kim [4], and Cho–Kim [5] for the isentropic case. Same to the non-vacuum case, the global well-posedness in 1D also holds for the vacuum case, for arbitrary large initial data, see the recent work by the author [27]. Generally, one can only expect the solutions in the homogeneous Sobolev spaces, see Li–Wang–Xin [26]. Global existence of strong solutions to the multi-dimensional compressible Navier-Stokes equations, with small initial data, in the presence of initial vacuum, was first proved by Huang–Li–Xin [17] for the isentropic case (see also Li–Xin [32] for further developments), and later by Huang–Li [16] and Wen–Zhu [48] for the non-isentropic case; in a recent work, the author [28] proved the global well-posedness result under the assumption that some scaling invariant quantity is small. Due to the finite blow-up results in [49, 50], the global solutions obtained in [16, 48, 28] must have unbounded entropy if the initial density is compactly supported; however, if the initial density has vacuum at the far field only, one can expect the global entropy-bounded solutions, see the recent work by the author and Xin [30, 31].

In all the global well-posedness results [24, 23, 52, 53, 1, 21, 29], for the heat conductive compressible Navier-Stokes equations in 1D, the density was assumed uniformly away from vacuum. For the vacuum case, global well-posedness of heat conductive compressible Navier-Stokes equations in 1D was proved by Wen-Zhu [47] with the heat conductive coefficient $\kappa\approx 1+\theta^{q}$, for positive $q$ suitably large, and by the author [27] with $\kappa\equiv Const.$

The aim of this paper is to study the global well-posedness of strong solutions to the one-dimensional non-heat conductive compressible Navier-Stokes equations, i.e., system (1.1)–(1.3), with constant viscosity, in the presence of vacuum; this is the counterpart of the paper [27] where the heat conductive case was considered. To our best knowledge, global well-posedness of 1D non-heat conductive compressible Navier-Stokes equations for arbitrary large initial data is not known before, no matter the vacuum is contained or not.

The results of this paper will be proven in the Lagrangian flow map coordinate being stated in the next subsection; however, it can be equivalently translated back to the corresponding one in the Euler coordinate.

1.2. The Lagrangian coordinates and main result

Let $\eta(y,t)$ be the flow map governed by $u$, that is

[TABLE]

Denote by $\varrho,v$, and $\pi$ the density, velocity, and pressure, respectively, in the Lagrangian coordinate, that is

[TABLE]

and introduce a function $J=J(y,t)=\eta_{y}(y,t)$. Then, it follows

[TABLE]

and system (1.4)–(1.6) can be rewritten in the Lagrangian coordinate as

[TABLE]

Due to (1.7) and (1.8), it is straightforward that

[TABLE]

from which, by setting $\varrho|_{t=0}=\varrho_{0}$ and noticing that $J|_{t=0}=1$, we have $J\varrho=\varrho_{0}.$ Therefore, one can replace (1.8) with (1.7), by setting $\varrho=\frac{\varrho_{0}}{J}$, and rewrite (1.9) as

[TABLE]

In summary, we only need to consider the following system

[TABLE]

We consider the initial-boundary value problem on the interval $(0,L)$, with $L>0$, and the boundary and initial conditions read as

[TABLE]

and

[TABLE]

We point out that here we put the initial condition on $\varrho_{0}v$ rather than on $v$. As will be shown in Theorem 1.1, in the below, we can guarantee the continuity in time of $\varrho_{0}v$ but not necessary of $v$, if the initial data lies only in $H^{1}$.

For $1\leq q\leq\infty$ and positive integer $m$, we use $L^{q}=L^{q}((0,L))$ and $W^{m,q}=W^{m,q}((0,L))$ to denote the standard Lebesgue and Sobolev spaces, respectively, and in the case that $q=2$, we use $H^{m}$ instead of $W^{m,2}$. $H_{0}^{1}$ consists of all functions $v\in H^{1}$ satisfying $v(0)=v(L)=0$. We always use $\|u\|_{q}$ to denote the $L^{q}$ norm of $u$.

The main result of this paper is the following:

Theorem 1.1**.**

Assume $0\leq\varrho_{0},\pi_{0}\in L^{\infty},$ and $v_{0}\in H_{0}^{1}.$ Then, there is a unique global solution $(J,v,\pi)$ to system (1.11)–(1.13), subject to (1.14)–(1.15), satisfying

[TABLE]

for any $T\in(0,\infty)$.

Remark 1.1**.**

The arguments presented in this paper also work for the free boundary value problem in which the boundary condition for $v$ in (1.14) is replaced by

[TABLE]

In fact, all the energy estimates obtained in this paper hold if replacing the boundary condition (1.14) with the above one, by copying or slightly modifying the proof.

Throughout this paper, we use $C$ to denote a general positive constant which may different from line to line.

2. Local and global well-posedness: without vacuum

This section is devoted to establishing the global well-posedness in the absence of vacuum which will be the base to prove the corresponding result in the presence of vacuum in the next section.

We start with the following local existence result of which the proof will be given in the appendix.

Proposition 2.1**.**

Assume that $(\varrho_{0},J_{0},v_{0},\pi_{0})$ satisfies

[TABLE]

for positive numbers $\underline{\varrho},\bar{\varrho},\underline{J},$ and $\bar{J}$.

Then, there is a positive time $T_{0}$ depending only on $R$, $\gamma$, $\mu$, $\underline{\varrho}$, $\bar{\varrho}$, $\underline{J}$, $\overline{J},$ and $\|(J_{0},v_{0},\pi_{0})\|_{H^{1}}$, such that system (1.11)–(1.13), subject to (1.14)–(1.15), has a unique solution $(J,v,\pi)$ on $(0,L)\times(0,T_{0})$, satisfying

[TABLE]

In the rest of this section, we always assume that $(J,v,\pi)$ is a solution to system (1.11)–(1.13), subject to (1.14)–(1.15), on $(0,L)\times(0,T)$, satisfying the regularities stated in Proposition 2.1, with $T_{0}$ there replaced by some positive time $T$. A series of a priori estimates of $(J,v,\pi)$, independent of the lower bound of the density, are carried out in this section.

We start with the basic energy identity.

Proposition 2.2**.**

It holds that

[TABLE]

and

[TABLE]

for any $t\in(0,\infty)$, where $\ell_{0}:=\int_{0}^{L}J_{0}dy$ and $E_{0}:=\int_{0}^{L}\left(\frac{\varrho_{0}}{2}v_{0}^{2}+\frac{\pi_{0}}{\gamma-1}\right)dy$.

Proof.

The first conclusion follows directly from integrating (1.11) with respect to $y$ over $(0,L)$ and using the boundary condition (1.14). Multiplying equation (1.12) by $v$, integrating the resultant over $(0,L)$, one gets from integrating by parts that

[TABLE]

Multiplying (1.13) with $J$ and integrating the resultant over $(0,L)$, it follows from (1.11) that

[TABLE]

which, combined with the previous equality, leads to

[TABLE]

the second conclusion follows. ∎

Next, we carry out the estimate on the lower bound of $J$. To this end, we perform some calculations in the spirit of [24] as preparations.

Due to (1.11), it follows from (1.12) that

[TABLE]

Integrating the above equation with respect to $t$ over $(0,t)$ yields

[TABLE]

from which, integrating with respect to $y$ over $(z,y)$, one obtains

[TABLE]

Thanks to this, noticing that

[TABLE]

and rearranging the terms, one obtains

[TABLE]

Therefore, both sides of the above equality are independent of the spacial variable, that is

[TABLE]

for some function $h$, from which, one can easily get

[TABLE]

where

[TABLE]

Multiplying both sides of the above with $\pi$ leads to

[TABLE]

from which, integrating with respect to $t$, one arrives at

[TABLE]

Thanks to the above, one can obtain from (2.16) that

[TABLE]

A prior positive lower bound of $J$ is stated in the following proposition:

Proposition 2.3**.**

The following estimate holds

[TABLE]

for any $t\in[0,\infty)$.

Proof.

By Proposition 2.2, it follows from the Hölder inequality that

[TABLE]

where $m_{0}=\int_{0}^{L}\varrho_{0}dy$, and, thus,

[TABLE]

Applying Proposition 2.2, using (2.18), and integrating (2.17) over $(0,L)$, one deduces

[TABLE]

and, thus,

[TABLE]

Applying the Gronwall inequality to the above yields

[TABLE]

With the aid of this and recalling $\pi\geq 0$ and (2.18), one obtains from (2.16) that

[TABLE]

the conclusion follows. ∎

Before continuing the argument, let us introduce the key quantity of this paper, the effective viscous flux $G$, defined as

[TABLE]

By some straightforward calculations, one can easily derive the equation for $G$ from (1.11)–(1.13) as

[TABLE]

Moreover, noticing that $\varrho_{0}v_{t}=G_{y}$, it is clear from the boundary condition of $v$, i.e., (1.14), that

[TABLE]

The next proposition concerning the estimate on $G$ is the key of proving the $H^{1}$ estimates on $(J,v,\pi)$ later.

Proposition 2.4**.**

The following estimate holds

[TABLE]

where $G_{0}=\mu\frac{v_{0}^{\prime}}{J_{0}}-\pi_{0}$ and $C$ depends only on $\gamma,\mu,\bar{\varrho},\ell_{0},\underline{J},m_{0},E_{0},$ and $T$.

Proof.

Multiplying (2.20) with $JG$ and recalling the boundary condition (2.21), it follows from integration by parts and (1.11) that

[TABLE]

Integration by parts and the Hölder inequality yield

[TABLE]

By the Gagliardo-Nirenberg inequality and applying Proposition 2.3, it follows

[TABLE]

Combining the previous two inequalities, it follows from the Young inequality and Proposition 2.2 that

[TABLE]

for any positive $\varepsilon$. Substituting the above into (2.22) with suitably chosen $\varepsilon$, one obtains

[TABLE]

which leads to the conclusion by applying the Gronwall inequality and simply using (2.23) and Proposition 2.3. ∎

The uniform upper bounds of $J,\pi$ can now be proved as in the next proposition.

Proposition 2.5**.**

The following estimate holds

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,\bar{\varrho},\ell_{0},\underline{J},m_{0},E_{0},\|G_{0}\|_{2}$, and $T$.

Proof.

Noticing that $v_{y}=\frac{J}{\mu}(G+\pi)$, one can rewrite (1.13) as

[TABLE]

from which one can further derive

[TABLE]

The estimate for $\pi$ follows straightforwardly from integrating (2.25) with respect to $t$ and applying Proposition 2.4. As for the estimate for $J$, noticing that (1.11) can be rewritten in terms of $G$ and $\pi$ as $J_{t}=\frac{J}{\mu}(G+\pi),$ the conclusion follows from the Gronwall inequality by Proposition 2.3, Proposition 2.4, and the estimate for $\pi$ just proved. ∎

A priori $L^{\infty}(0,T;H^{1})$ estimate for $(J,\pi)$ is given in the next proposition.

Proposition 2.6**.**

The following estimate holds

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,\bar{\varrho},\ell_{0},\underline{J},\bar{J},m_{0},\|\pi_{0}\|_{\infty},E_{0},\|G_{0}\|_{2}$, and $T$.

Proof.

Differentiating (2.24) with respect to $y$ gives

[TABLE]

Multiplying the above equation with $\pi_{y}$ and integrating over $(0,L)$, one deduces

[TABLE]

and, thus, by the Gronwall inequality, and applying Proposition 2.4 and Proposition 2.5, one gets

[TABLE]

Note that

[TABLE]

Therefore, by Proposition 2.4 and the estimate just obtained for $\|\pi_{y}\|_{2}$, it follows that

[TABLE]

and further by Proposition 2.5 that

[TABLE]

proving the conclusion. ∎

Corollary 2.1**.**

It holds that

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,\bar{\varrho},\ell_{0},\underline{J},\bar{J},m_{0},\|\pi_{0}\|_{\infty},E_{0}$, $\|G_{0}\|_{2}$, $\|J_{0}^{\prime}\|_{2}$, $\|\pi_{0}^{\prime}\|_{2}$, and $T$.

Proof.

The estimates on $\sup_{0\leq t\leq T}\|v_{y}\|_{2}^{2}$ and $\int_{0}^{T}\|\sqrt{\varrho_{0}}v_{t}\|_{2}^{2}dt$ follow from Propositions 2.4 and 2.5 by noticing that $v_{y}=\frac{J}{\mu}(G+\pi)$ and $\sqrt{\varrho_{0}}v_{t}=\frac{G_{y}}{\sqrt{\varrho_{0}}}$. As for the estimate of $v_{yy}$, noticing that

[TABLE]

it follows from Propositions 2.4–2.6 that

[TABLE]

The estimate for $J_{t}$ follows directly from (1.11) and the estimates obtained. By Propositions 2.4–2.6, it follows from (2.25) that

[TABLE]

and

[TABLE]

This completes the proof. ∎

The following $t$-weighted estimates will be used in the compactness arguments in the passage of taking limit from the non-vacuum case to the vacuum case.

Proposition 2.7**.**

The following estimate holds

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,\bar{\varrho},\ell_{0},\underline{J},\bar{J},m_{0},\|\pi_{0}\|_{\infty},E_{0}$, $\|G_{0}\|_{2}$, $\|J_{0}^{\prime}\|_{2}$, $\|\pi_{0}^{\prime}\|_{2}$, and $T$.

Proof.

Multiplying (2.20) with $JG_{t}$, then integrating by parts yields

[TABLE]

which, multiplied with $t$, gives

[TABLE]

Integrating the above with respect to $t$, and using Proposition 2.4 and Corollary 2.1 yield

[TABLE]

Recalling the expression of $G$, by direct calculations, and using (1.13), one deduces

[TABLE]

which gives

[TABLE]

Therefore, it follows from (2.26), Proposition 2.4, and Corollary 2.1 that

[TABLE]

proving the conclusion. ∎

In summary, we have the following

Corollary 2.2**.**

The following estimates hold

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,\bar{\varrho},\underline{J},\|(J_{0},v_{0},\pi_{0})\|_{H^{1}},$ and $T$.

Proof.

This is a direct corollary of Propositions 2.3, 2.5, 2.6, 2.7, and Corollary 2.1, by using some necessary embedding inequalities. ∎

Remark 2.1**.**

Checking the proofs of Propositions 2.4–2.7, one can easily see that all the constants $C$ in the arguments viewing as functions of $T$ can be chosen in such a way that are continuous in $T\in[0,\infty)$.

We conclude this section with the following global well-posedness result for the non-vacuum case.

Theorem 2.1**.**

Under the conditions in Proposition 2.1, there is a unique global solution $(J,v,\pi)$ to system (1.11)–(1.13), subject to (1.14)–(1.15), satisfying

[TABLE]

Proof.

By Proposition 2.1, there is a unique local solution $(J,v,\pi)$ to system (1.11)–(1.13), subject to (1.14)–(1.15). By iteratively applying Proposition 2.1, one can extend the local solution to the maximal time of existence $T_{\text{max}}$. We claim that $T_{\text{max}}=\infty$. Assume by contradiction that $T_{\text{max}}<\infty$. Then, by Corollary 2.2 and recalling Remark 2.1, there is a positive constant $C$, independent of $T\in(0,T_{\text{max}})$, such that

[TABLE]

Thanks to this, by the local existence result, Proposition 2.1, one can extend the local solution $(J,v,\pi)$ beyond $T_{\text{max}}$, contradicting to the definition of $T_{\text{max}}$. Therefore, it must have $T_{\text{max}}=\infty$. This proves the conclusion. ∎

3. Global well-posedness: in the presence of vacuum

In this section, we prove our main result as follows.

Proof of Theorem 1.1.

**Existence. **Choose $\varrho_{0n}\in H^{1}$, with $\frac{1}{n}\leq\varrho_{0n}\leq\bar{\varrho}+1$, such that $\varrho_{0n}\rightarrow\varrho_{0}$ in $L^{q}$, for any $q\in(1,\infty)$. By Theorem 2.1, for any $n$, there is a unique global solution $(J_{n},v_{n},\pi_{n})$ to system (1.11)–(1.13), subject to (1.14)–(1.15), with $\varrho_{0}$ in (1.12) replaced with $\varrho_{0n}$. By Corollary 2.2, there is a positive constant $C$, independent of $n$, such that

[TABLE]

for any $T\in(0,\infty)$. By the Aubin-Lions lemma, and using Cantor’s diagonal argument, there is a subsequence, still denoted by $(J_{n},v_{n},\pi_{n})$, and $(J,v,\pi)$ enjoying the regularities

[TABLE]

such that

[TABLE]

and

[TABLE]

Here, $\rightarrow$, $\rightharpoonup$, and $\overset{*}{\rightharpoonup}$ denote, respectively, the strong, weak, and weak* convergence in the corresponding spaces. Thanks to (3.31)–(3.37), one can take the limit $n\rightarrow\infty$ to show that $(J,v,\pi)$ is a solution to system (1.11)–(1.13), on $(0,L)\times(0,T)$. Moreover, recalling $(J_{n},\pi_{n})|_{t=0}=(J_{0},\pi_{0})$, it is clear from (3.36) and (3.38) that $(J,\pi)|_{t=0}=(J_{0},\pi_{0})$.

One needs to verify the regularities of $(J,v,\pi)$ and that $(\varrho_{0}v)|_{t=0}=\varrho_{0}v_{0}$. Using (3.27) and (3.34), by the lower semi-continuity of the norms, one deduces

[TABLE]

for any $\delta\in(0,T)$, and for a positive constant $C$ independent of $\delta$, and, thus, $\sqrt{\varrho_{0}}v_{t}\in L^{2}(0,T;L^{2}).$ The desired regularities $J,\pi\in C([0,T];H^{1})$ follow from (3.28) and (3.30).

It remains to verify $\varrho_{0}v\in C([0,T];L^{2})$ and $(\varrho_{0}v)|_{t=0}=\varrho_{0}v_{0}.$ To this end, noticing that (3.29) and (3.30) imply $v\in C((0,T];H^{1})$, it suffices to show that $(\varrho_{0}v)(\cdot,t)\rightarrow\varrho_{0}v_{0}$, strongly in $L^{2}$, as $t\rightarrow 0$. Using (3.27), it follows

[TABLE]

for a positive constant $C$ independent of $n$. Recalling (3.37) and $\varrho_{0n}\rightarrow\varrho_{0}$, for any $q>1$, one has

[TABLE]

It follows from (3.39) that

[TABLE]

where $C$ is independent of $n$, from which, recalling (3.40), one can take the limit $n\rightarrow\infty$ to get

[TABLE]

This proves the continuity of $\varrho_{0}v$ at $t=0$ and verifies $\varrho_{0}v|_{t=0}=\varrho_{0}v_{0}.$

Therefore, $(J,v,\pi)$ is a global solution to system (1.11)–(1.13), subject to the initial and boundary conditions (1.14)–(1.15), satisfying the regularities stated in Theorem 1.1. This proves the existence part of Theorem 1.1.

Uniqueness. Let $(J_{1},v_{1},\pi_{1})$ and $(J_{2},v_{2},\pi_{2})$ be two solutions to system (1.11)–(1.13), subject to (1.14)–(1.15), and denote $(J,v,\pi):=(J_{1}-J_{2},v_{1}-v_{2},\pi_{1}-\pi_{2})$. Then, straightforward calculations lead to

[TABLE]

Multiplying (3.41), (3.42), and (3.43), respectively, with $J,v,$ and $\pi$, and integrating the resultants over $(0,L)$, one gets from integration by parts and using the Young inequalities that

[TABLE]

where the fact that $J_{1}$ and $J_{2}$ have positive lower bounds on $(0,L)\times(0,T)$ for any finite $T$ has been used. Adding up the previous three inequalities and choosing $\varepsilon$ sufficiently small, one obtains

[TABLE]

from which, noticing that $\pi_{i},v_{iy}\in L^{2}(0,T;L^{\infty}),i=1,2$, and by the Gronwall inequality, one obtains $J\equiv\pi\equiv\sqrt{\varrho_{0}}v\equiv v_{y}\equiv 0$. Thanks to this, by the Poincaré inequality, the uniqueness follows. ∎

4. Appendix: local well-posedness, i.e., proof of Proposition 2.1

In this appendix, we prove the local well-posedness of system (1.11)–(1.13), subject to (1.14)–(1.15), for the case that the the initial density $\varrho_{0}$ is uniformly away from vacuum. In other words, we give the proof of Proposition 2.1.

For positive time $T\in(0,\infty)$, denote

[TABLE]

and

[TABLE]

For positive numbers $M$ and $T$, we denote

[TABLE]

By the Poincaré inequality, one can verify that $\mathscr{K}_{M,T}$ is a closed subset of $X_{T}$.

Given $(\varrho_{0},J_{0},v_{0},\pi_{0})$, satisfying

[TABLE]

for positive numbers $\underline{\varrho},\bar{\varrho},\underline{J},$ and $\bar{J}$.

Define three mappings $\mathscr{Q},\mathscr{R},$ and $\mathscr{F}$ as follows. First, for $v\in\mathscr{K}_{M,T}$, define $J=\mathscr{Q}(v)$ as the unique solution to

[TABLE]

Next, for given $v\in\mathscr{K}_{M,T}$, and with $J$ solved as above, define $\pi=\mathscr{R}(v)$ as the unique solution to

[TABLE]

And finally, for given $v\in\mathscr{K}_{M,T}$, and with $J$ and $\pi$ solved as above, define $V=\mathscr{F}(v)$ as the unique solution to

[TABLE]

It is clear that

[TABLE]

where

[TABLE]

In order to prove the local existence and uniqueness of solutions to system (1.11)–(1.13), subject to (1.14)–(1.15), and recalling the definitions of the mappings $\mathscr{Q},\mathscr{R}$, and $\mathscr{F}$, it suffices to show that the mapping $\mathscr{F}$ has a unique fixed point in $X_{T}$, which will be proved by the contractive mapping principle.

For simplicity of notations, throughout this section, we agree the following:

[TABLE]

for arbitrary $v,v_{1},v_{2}\in\mathscr{K}_{M,T}$. By the Poincáre and Gagliardo-Nirenberg inequality, there is a positive constant $C_{1}$ depending only on $L$, such that

[TABLE]

This kind inequality for $v$ will be frequently used without further mentions, and we use $C_{1}$ specifically to denote the constant in the above inequality.

In the rest of this section, we always assume that $M$ and $T$ are two positive constants, to be determined later, satisfying

[TABLE]

Proposition 4.1**.**

(i) For any $v\in X_{T}$, it follows that

[TABLE]

(ii) Consequently, for any $v\in\mathscr{K}_{M,T}$, one has

[TABLE]

Proof.

For any $v\in X_{T}$, by the Hölder inequality and (4.47), one deduces

[TABLE]

which leads to the first inequality in (i). The second inequality in (i) follows from the first one by simply applying the Hölder inequality. The inequalities in (ii) follow from those in (i) by using the conditions in (4.48). ∎

4.1. Properties of $\mathscr{Q}$

Proposition 4.2**.**

(i) It holds that

[TABLE]

for any $v,v_{1},v_{2}\in\mathscr{K}_{M,T}$.

(ii) Assume, in addition, that $T\leq\left(\frac{\underline{J}}{2C_{1}}\right)^{2}$. Then,

[TABLE]

for any $v\in\mathscr{K}_{M,T}$.

Proof.

(i) Recalling the expression of $\mathscr{Q}$, it is clear that

[TABLE]

where (4.48) has been used. Similarly,

[TABLE]

By (i) of Proposition 4.1, one deduces

[TABLE]

(ii) If $T\leq(\frac{\underline{J}}{2C_{1}})^{2}$, it follows from (ii) of Proposition 4.1 that

[TABLE]

and, consequently,

[TABLE]

proving the conclusion. ∎

Due to Proposition 4.2, in the rest of this section, we always assume, in addition to (4.48), that $T\leq\left(\frac{\underline{J}}{2c_{1}}\right)^{2}$, so that (ii) of Proposition 4.2 applies.

Proposition 4.3**.**

The following estimates hold:

[TABLE]

and

[TABLE]

for any $0\leq\tau<t\leq T$, and $v_{1},v_{2}\in\mathscr{K}_{M,T}$, where $C$ is a positive constant depending only on $\gamma,L,\underline{J}$.

Proof.

Applying Proposition 4.1 and Proposition 4.2, one deduces

[TABLE]

By the mean value theorem, there is a number $\eta\in(0,1)$, such that

[TABLE]

Thus, using (4.49), it follows from Proposition 4.1 and Proposition 4.2 that

[TABLE]

proving the conclusion. ∎

Proposition 4.4**.**

The following estimates hold

[TABLE]

and

[TABLE]

for any $0\leq\tau<t\leq T$, and for any $v,v_{1},v_{2}\in\mathscr{K}_{M,T}$, where $C$ is a positive constant depending only on $\gamma,L,$ and $\underline{J}$.

Proof.

Note that $(\frac{v_{y}}{J})_{y}=\frac{v_{yy}}{J}-\frac{J_{y}v_{y}}{J^{2}}$, it follows from Proposition 4.1 and Proposition 4.2 that

[TABLE]

The $L^{1}(0,T;L^{2})$ estimate for $(\frac{v_{y}}{\mathcal{Q}(v)})_{y}$ follows from the above inequality by simply using the Hölder inequality.

For simplicity of notations, for $v_{1},v_{2}\in\mathscr{K}_{M,T}$, we denote $\delta v=v_{1}-v_{2}$, $J_{i}=\mathscr{Q}(v_{i})$, $i=1,2$, and $\delta J=J_{1}-J_{2}$. By direct calculations

[TABLE]

Therefore, it follows from Proposition 4.1 and Proposition 4.2 that

[TABLE]

Straightforward computations yield

[TABLE]

Therefore, it follows from Propositions 4.1, 4.3, and 4.4 that

[TABLE]

proving the conclusion. ∎

4.2. Properties of $\mathscr{R}$

Proposition 4.5**.**

It holds that

[TABLE]

for any $v_{1},v_{2}\in\mathscr{K}_{M,T}$, and for a positive constant $C$ depending only on $\gamma$, $L$, $\underline{J}$, $\|J_{0}^{\prime}\|_{2}$, $\|\pi_{0}\|_{\infty},$ and $\|\pi_{0}^{\prime}\|_{2}$.

Proof.

For simplicity of notations, for $v_{1},v_{2}\in\mathscr{K}_{M,T}$, we denote $\delta v=v_{1}-v_{2}$, $J_{i}=\mathscr{Q}(v_{i})$, $i=1,2$, and $\delta J=J_{1}-J_{2}$. Note that

[TABLE]

It follows from Proposition 4.3 and Proposition 4.4 that

[TABLE]

proving the conclusion. ∎

Proposition 4.6**.**

It holds that

[TABLE]

for any $v_{1},v_{2}\in\mathscr{K}_{M,T}$, and for a positive constant $C$ depending only on $\gamma,L,\underline{J},$ and $\|J_{0}^{\prime}\|_{2}$.

Proof.

For simplicity of notations, for $v_{1},v_{2}\in\mathscr{K}_{M,T}$, we denote $\delta v=v_{1}-v_{2}$, $J_{i}=\mathscr{Q}(v_{i})$, $i=1,2$, and $\delta J=J_{1}-J_{2}$. Straightforward calculations yield

[TABLE]

Estimates for $I_{i},i=1,2,3,4,5$, are give as follows. By Proposition 4.1 and Proposition 4.4

[TABLE]

Similarly, it follows from Propositions 4.1, 4.3, and 4.4 that

[TABLE]

where we have used (4.48). It follows from Propositions 4.1–4.4 that

[TABLE]

where (4.48) has been used. By Propositions 4.1–4.4, one deduces

[TABLE]

and

[TABLE]

where $MT^{\frac{1}{4}}$ has been used. Therefore, we have

[TABLE]

proving the conclusion. ∎

Proposition 4.7**.**

For any $v\in\mathscr{K}_{M,T}$, it holds that

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,L,\underline{J},\|J_{0}^{\prime}\|_{2},\|\pi_{0}\|_{\infty},$ and $\|\pi_{0}^{\prime}\|_{2}$.

Proof.

Note that $\mathscr{R}(0)=\pi_{0}$, it follows from Proposition 4.5 and Proposition 4.6 that

[TABLE]

where (4.48) has been used. This proves the conclusion. ∎

4.3. Properties of $\mathscr{F}$

Proposition 4.8**.**

For any $v\in\mathscr{K}_{M,T}$, it holds that

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,L,\underline{\varrho},\bar{\varrho},\underline{J},\|J_{0}^{\prime}\|_{2},\|\pi_{0}\|_{\infty},\|\pi_{0}^{\prime}\|_{2}$, and $\|v_{0}^{\prime}\|_{2}$.

Proof.

Denote $J=\mathscr{Q}(v),\pi=\mathscr{R}(v)$, and $V=\mathscr{F}(v)$. Testing (4.46) with $-\frac{V_{yy}}{\varrho_{0}}$ and noticing $\frac{\underline{J}}{2}\leq J\leq 2\bar{J}$, one deduces

[TABLE]

for any positive $\varepsilon$, which, choosing $\varepsilon$ sufficiently small and applying Propositions 4.1, 4.2, and 4.7, gives

[TABLE]

proving the conclusion. ∎

Proposition 4.9**.**

It holds that

[TABLE]

for a positive constant $C$ depending only on $\gamma,\mu,L,\underline{\varrho},\bar{\varrho},\underline{J},\|J_{0}^{\prime}\|_{2},\|\pi_{0}\|_{\infty},\|\pi_{0}^{\prime}\|_{2}$, and $\|v_{0}^{\prime}\|_{2}$.

Proof.

Denote $J_{i}=\mathscr{Q}(v_{i}),\pi_{i}=\mathscr{R}(v_{i}),V_{i}=\mathscr{F}v_{i},i=1,2.$ Set $\delta J=J_{1}-J_{2},\delta\pi=\pi_{1}-\pi_{2},$ and $\delta V=V_{1}-V_{2}$. Then,

[TABLE]

Testing the above with $-\delta V_{yy}$ and using Proposition 4.2, one deduces

[TABLE]

which, integrating with respect to $t$, and applying Propositions 4.1, 4.2, 4.5, 4.6, and 4.8, yields

[TABLE]

proving the conclusion. ∎

Corollary 4.1**.**

There is a positive constant $C_{\#}$ depending only on $\gamma,\mu,L,\underline{\varrho}$, $\bar{\varrho}$, $\underline{J},\|J_{0}^{\prime}\|_{2},\|\pi_{0}\|_{\infty},\|\pi_{0}^{\prime}\|_{2}$, and $\|v_{0}^{\prime}\|_{2}$, such that for any $M\geq C_{\#}$, it follows

[TABLE]

for any $v,v_{1},v_{2}\in\mathscr{K}_{M,T_{\#}}$, where

[TABLE]

Proof.

By Proposition 4.8 and Proposition 4.9, there is a positive constant $C_{\#}$ depending only on $\gamma,\mu,L,\underline{\varrho},\bar{\varrho},\underline{J},\|J_{0}^{\prime}\|_{2},\|\pi_{0}\|_{\infty},\|\pi_{0}^{\prime}\|_{2}$, and $\|v_{0}^{\prime}\|_{2}$, such that

[TABLE]

for any $v,v_{1},v_{2}\in\mathscr{K}_{M,T}$, for any $M,T$ satisfying

[TABLE]

For any $M\geq C_{\#}$, choose

[TABLE]

Then, by (4.51), one has

[TABLE]

for any $v,v_{1},v_{2}\in\mathscr{K}_{M,T_{\#}}$, proving the conclusion. ∎

4.4. Properties of $\mathscr{F}$ and the local well-posedness

Proof of Proposition 2.1.

Let $C_{\#}$ be the positive constant in Corollary 4.1. Set $M=C_{\#}$ and let $T_{\#}$ be the corresponding positive time in Corollary 4.1. Recall the definition of $\mathscr{K}_{M_{\#},T_{\#}}$ and define $|||v|||:=\|v_{y}\|_{V_{T}}$, for any $v\in\mathscr{K}_{M_{\#},T_{\#}}$. By the Poincaré inequality, one can easily check that $|||\cdot|||$ is a norm on the space $X_{T_{\#}}$ and is equivalent to the $L^{\infty}(0,T_{\#};H_{0}^{1})\cap L^{2}(0,T_{\#};H^{2})$ norm. Consequently, $\mathscr{K}_{M_{\#},T_{\#}}$ is a completed metric space, equipped with the metric $d(v_{1},v_{2}):=|||v_{1}-v_{2}|||=\|\partial_{y}(v_{1}-v_{2})\|_{V_{T}}$. Let $\mathscr{Q},\mathscr{R},\mathscr{F}$ be the mappings defined as before. By Corollary 4.1, $\mathscr{F}$ is a contractive mapping on $\mathscr{K}_{M_{\#},T_{\#}}$. Therefore, by the contractive mapping principle, there is a unique fixed point, denoted by $v_{\#}$, to $\mathscr{F}$ on $\mathscr{K}_{M_{\#},T_{\#}}$. Set $J_{\#}=\mathscr{Q}(v_{\#})$ and $\pi_{\#}=\mathscr{R}(v_{\#})$. By the definitions of $\mathscr{Q}(v_{\#})$ and $\mathscr{R}(v_{\#})$, one can easily check that $(J_{\#},v_{\#},\pi_{\#})$ is a solution to system (1.11)–(1.13), subject to (1.14)–(1.15). The regularities of $(J_{\#},v_{\#},\pi_{\#})$ can be verified through straightforward computations to the expressions of $\mathscr{Q}(v)$ and $\mathscr{R}(v)$ and using (4.46). Since the calculations are standard, we omit the details here. ∎

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China NSFC 11971009, NSFC 11771156, and NSFC 11871005, the start-up grant of the South China Normal University 550-8S0315, and the Hong Kong RGC grant CUHK 14302917.