Asymptotic stability of a shock wave for compressible non-miscible two-phase flows in 1D

AoMing ZHAO; YaZhou CHEN

doi:10.13543/j.bhxbzr.2024.05.015

Journal of Beijing University of Chemical Technology >

2024 , Vol. 51 >Issue 5: 121 - 128

DOI: https://doi.org/10.13543/j.bhxbzr.2024.05.015

Management and Mathematics

Asymptotic stability of a shock wave for compressible non-miscible two-phase flows in 1D

AoMing ZHAO ,
YaZhou CHEN ^,*

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College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

Received date: 2023-03-13

Online published: 2024-10-15

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Abstract

In this paper, the large-time behavior of the solution to the Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system, which describes the flow of non-miscible two-phase flows with diffusion interfaces, has been studied. Using the anti-derivative and energy method, we demonstrate the existence and asymptotic stability of the viscous shock solution for one-dimensional compressible Navier-Stokes/Allen-Cahn equation.

Key words： Navier-Stokes/Allen-Cahn system; shock wave; asymptotic stability

Cite this article

AoMing ZHAO , YaZhou CHEN . Asymptotic stability of a shock wave for compressible non-miscible two-phase flows in 1D[J]. Journal of Beijing University of Chemical Technology, 2024 , 51(5) : 121 -128 . DOI: 10.13543/j.bhxbzr.2024.05.015

引言

非混相两相流在自然界十分常见，在工业领域如水利工程、热能工程以及化工冶金等方面也有着重要的应用价值。

有关两相流理论的研究前人已经有很多成果。Blesgen^[1]在1999年首先提出了可压缩Navier-Stokes/Allen-Cahn（NSAC）模型。当相场为常数时，NSAC方程变成经典的Navier-Stokes（NS）方程，其无黏情况对应于Euler方程。对于一维等熵的Euler方程组基本波问题，Oleinik等^[2]于20世纪60年代开创性地使用极值原理得到了单个黏性守恒律方程激波和稀疏波的稳定性。如果考虑NS方程的基本波问题，对于一维可压缩等熵的NS方程，Matsumura等^[3]和Goodman^[4]首先研究了在小扰动条件下黏性激波的稳定性。此外，对于一维条件下可压缩非等熵NS方程的Cauchy问题，若考虑系统为理想多方气体，且热传导系数为常数的情况，Kawashima等^[5]拓展了文献[3]的结果，得到了非等熵可压缩黏性激波的渐近稳定性。对于一维可压缩NSAC方程的初边值问题，Ding等^[6]得到了不包含真空时整体解的存在唯一性，Chen等^[7]得到了包含真空时整体解的存在唯一性，孙颖等^[8]得到了一类具有周期边值条件的NSAC方程整体解的存在唯一性。对于一维NSAC方程的Cauchy问题，当考虑相场在无穷远处为1时，Luo等^[9-10]得到了等熵和非等熵系统的稀疏波解的存在性和大时间行为。进一步地，Luo等^[11]还得到了NSAC方程一组复合波（2-稀疏波和接触间断）解的大时间行为。Chen等^[12]得到了可压缩NSAC方程尖锐界面极限。

在前人工作的基础上，本文考虑初值在相分离状态附近（即相场在无穷处为±1）的情形，研究一维可压缩NSAC方程组黏性激波解的存在性和渐近稳定性。此外，为了相对容易地得到比容（密度的倒数）的高阶估计，并克服相场和密度耦合项带来的计算困难，我们借鉴了Vasseur等^[13]使用的有效速度方法，并降低了初始条件的正则性。

1 模型的构造及主要定理

本文考虑一维等熵可压缩NSAC方程组^[6]，并利用Langrange坐标变换，可以得到如下系统。

(1)

$\left\{\begin{array}{l}v_{\tilde{t}}-u_{\tilde{x}}=0 \\u_{\tilde{t}}+p_{\tilde{x}}(v)=\left(\frac{u_{\tilde{x}}}{v}\right)_{\tilde{x}}-\frac{1}{2}\left(\frac{\chi_{\tilde{x}}^{2}}{v^{2}}\right)_{\tilde{x}} \\\chi_{\tilde{t}}=-v \mu \\\mu=\left(\chi^{3}-\chi\right)-\left(\frac{\chi_{\tilde{x}}}{v}\right)_{\tilde{x}}\end{array}\right.$

初值满足

(2)

$\begin{equation*}\left.(v, u, \chi)\right|_{\tilde{i}=0}=\left(v_{0}, u_{0}, \chi_{0}\right)(\tilde{x}) \xrightarrow{\tilde{x} \rightarrow \infty}\left(v_{ \pm}, u_{ \pm}, \pm 1\right) \end{equation*}$

式中，v、u、χ分别为气液混合物的比容（密度的倒数）、速度及组分的浓度差，p为压力项，μ为化学势能。

不失一般性，假设p(v)是v的函数，且满足

(3)

$p^{\prime}(v)<0, p^{\prime \prime}(v) \geqslant 0, p^{\prime \prime}(v) \not \equiv 0$

考虑如下NS方程。

(4)

$\left\{\begin{array}{l}v_{\tilde{t}}-u_{\tilde{x}}=0 \\u_{\tilde{t}}+p_{\tilde{x}}(v)=\left(\frac{u_{\tilde{x}}}{v}\right)_{\tilde{x}} \\(v, u)(\tilde{x}, 0)=\left(v_{0}, u_{0}\right)(\tilde{x})\end{array}\right.$

上述方程初值满足式（2）时，有黏性激波解

$(V, U)(\tilde{x}-s \tilde{t})$

满足如下方程^[3]。

(5)

$\left\{\begin{array}{l}-s V_{y}-U_{y}=0 \\-s U_{y}+p_{y}(V)=\left(\frac{U_{y}}{V}\right)_{y} \\\lim\limits_{y \rightarrow \pm \infty}(V, U)=\left(v_{ \pm}, u_{ \pm}\right)\end{array}\right.$

式中，

$t=\tilde{t}, y=\tilde{x}-s \tilde{t}, (V, U)(y)$

满足

(6)

$\left\{\begin{array}{l}V_{y}=\frac{V}{s}\left[p\left(v_{ \pm}\right)+s^{2}\left(v_{ \pm}-V\right)-p(V)\right] \\0<V_{y} \leqslant \frac{v_{+}}{s}\left(p\left(v_{-}\right)+s^{2} v_{-}\right) \\U_{y}=-s V_{y} \leqslant 0 \\v_{-}<V<v_{+}\end{array}\right.$

且存在正常数c_±满足

(7)

$\left\{\begin{array}{l}\left|(V, U)-\left(v_{ \pm}, u_{ \pm}\right)\right|=O(1) \delta_1 \mathrm{e}^{-c_{ \pm} \delta_1|y|} \\\left|V_y, U_y\right|=O(1) \delta_1^2 \mathrm{e}^{-c_{ \pm} \delta_1|y|}\end{array}\right.$

式中，

$\delta_{1}=v_{+}-v_{-} \mid, \left(v_{-}, u_{-}\right) 、\left(v_{+}, u_{+}\right)$

分别是激波的左状态和右状态，s > 0且满足以下R-H条件

$\left\{\begin{array}{l}s\left(v_{+}-v_{-}\right)=u_{-}-u_{+} \\s\left(u_{+}-u_{-}\right)=p\left(v_{+}\right)-p\left(v_{-}\right)\end{array}\right.$

下面引入反导数，设

(8)

$\left\{\begin{array}{l}\phi=\int_{-\infty}^{y}(v(z, t)-V(z+\alpha)) \mathrm{d} z \\\psi=\int_{-\infty}^{y}(u(z, t)-U(z+\alpha)) \mathrm{d} z\end{array}\right.$

令

(9)

$\left\{\begin{array}{l}\phi_{0}=\int_{-\infty}^{y}\left(v_{0}(z, t)-V(z+\alpha)\right) \mathrm{d} z \\ \psi_{0}=\int_{-\infty}^{y}\left(u_{0}(z, t)-U(z+\alpha)\right) \mathrm{d} z\end{array}\right.$

更进一步假设

(10)

$\left\{\begin{array}{l}\int_{-\infty}^{+\infty}\left(v_{0}(z, t)-V(z+\alpha)\right) \mathrm{d} z=0 \\\int_{-\infty}^{+\infty}\left(u_{0}(z, t)-U(z+\alpha)\right) \mathrm{d} z=0\end{array}\right.$

式中，

$\alpha=\frac{1}{v_{+}-v_{-}} \int_{-\infty}^{+\infty}\left(v_{0}(y)-V(y)\right) \mathrm{d} y$

。

本文中L²表示定义在

$\mathbb{R}$

上的可积函数组成的空间，其范数为

$\|f\|=\left(\int_{\mathbb{R}}|f|^{2} \mathrm{~d} y\right)^{1 / 2}, H^{l}(l \geqslant 0)$

表示Sobolev空间，其范数为

$\|f\|_{l}=\left(\sum\limits_{j=1}^{l}\left\|\partial_{y}^{j} f\right\|^{2}\right)^{1 / 2}$

。

定理1 假设初值满足

$\begin{aligned}& \left(\phi_{0}, \psi_{0}\right) \in H^{2}(\mathbb{R}), \chi_{0}^{2}-1 \in L^{2}(\mathbb{R}) \\& \chi_{0 \bar{x}} \in H^{2}(\mathbb{R}), \inf\limits_{\mathbb{R}} v_{0}>0\end{aligned}$

那么存在一个正常数δ，使得当

$\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}+\left\|\chi_{0 \dot{x}}\right\|_{2}+\left\|\chi_{0}^{2}-1\right\|+\left|v_{+}-v_{-}\right| \leqslant \delta$

时，Cauchy问题（1）、（2）有唯一的整体解(v, u, χ)满足

$\left\{\begin{array}{l}(v-V, u-U) \in C^{0}\left([0, +\infty) ; H^{1}(\mathrm{R})\right) \\\chi^{2}-1 \in C^{0}\left([0, +\infty) ; L^{2}(\mathrm{R})\right) \\\chi_{\tilde{x}} \in C^{0}\left([0, +\infty) ; H^{2}(\mathrm{R})\right) \cap L^{2}\left([0, +\infty) ; H^{3}(\mathrm{R})\right) \\v-V \in L^{2}\left([0, +\infty) ; H^{1}(\mathrm{R})\right) \\u-U \in L^{2}\left([0, +\infty) ; H^{2}(\mathrm{R})\right) \\-1 \leqslant \chi \leqslant 1\end{array}\right.$

此外，有如下大时间行为

$\lim\limits_{t \rightarrow+\infty}\left\|\left(v-V, u-U, \chi^{2}-1\right)\right\|_{L^{*}(\mathbb{R})}=0$

2 主要定理的证明

定理1的证明主要是利用Schauder不动点定理证明局部解的存在性，并运用反导数和能量估计方法得到该解的一致估计，从而将其延拓到全局解。

2.1 局部解的存在性

利用方程（1）、（5），结合反导数定义得到如下方程组。

(11)

$\left\{\begin{array}{l}\phi_{t}-s \phi_{y}-\psi_{y}=0 \\\psi_{t}-s \psi_{y}-f\left(V, U_{y}\right) \phi_{y}-\frac{\psi_{y y}}{V}=W-\frac{1}{2} \frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}} \\\chi_{t}-s \chi_{y}=-\left(V+\phi_{y}\right) \mu \\\mu=\left(\chi^{3}-\chi\right)-\left(\frac{\chi_{y}}{V+\phi_{y}}\right)_{y} \\(\phi, \psi, \chi)(y, 0)=\left(\phi_{0}, \psi_{0}, \chi_{0}\right)(y)\end{array}\right.$

其中

$f\left(V, U_{y}\right)=-p^{\prime}(V)-\frac{U_{y}}{V^{2}}>0, W=\frac{u_{y}}{v}-\frac{U_{y}}{V}-\\\frac{\psi_{y y}}{V}+\frac{U_{y} \phi_{y}}{V^{2}}-\left[p(v)-p(V)-p^{\prime}(V) \phi_{y}\right]$

对于任意给定的m, M > 0，定义如下解空间。

(12)

$\begin{align*}& X_{m, M}(0, T)=\left\{(\phi, \psi, \chi) \mid(\phi, \psi) \in C\left(0, T ; H^{2}\right), \right. \\& \chi^{2}-1 \in C\left(0, T ; L^{2}\right), \chi_{y} \in C\left(0, T ; H^{2}\right), \\& \sup\limits_{t \in(0, T)}\left(\|(\phi, \psi)\|_{2}+\left\|\chi_{y}\right\|_{2}+\left\|\chi^{2}(t)-1\right\|\right) \leqslant M, \\& \left.\inf\limits_{y \in \mathbb{R}_{, }, t \in(0, T)} v(y, t) \geqslant m>0\right\} \end{align*}$

命题1给出了方程（11）局部解的存在唯一性的结论。

命题1 ∀M > 0，假如初始值满足

(13)

$\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}+\left\|\chi_{0 y}\right\|_{2}+\left\|\chi_{0}^{2}-1\right\| \leqslant M$

(14)

$\begin{equation*}\inf\limits_{y \in \mathbb{R}}\left(V+\phi_{0, }\right)>m>0 \end{equation*}$

那么存在足够小的T^*，使得方程组（11）存在唯一的解

$(\phi, \psi, \chi) \in X_{\frac{m}{2}, 2 M}\left(\left[0, T^{*}\right]\right)$

，满足

$\|(\phi, \psi)\|_{2}^{2}+\left\|\chi^{2}-1\right\|^{2}+\left\|\chi_{y}\right\|_{2}^{2}+\int_{0}^{t}\left(\left\|\phi_{y}\right\|_{1}^{2}+\right.\\\left.\left\|\psi_{y}\right\|_{2}^{2}+\left\|\chi_{y}\right\|_{3}^{2}\right) \mathrm{d} \tau \leqslant C_{0}\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\right.\\\left.\left\|\chi_{0, }\right\|_{2}^{2}\right)$

并且-1≤χ≤1。

命题1可以由标准不动点定理证明，此处省去证明细节。为了得到全局解的存在性，将在命题2中给出先验估计。

2.2 全局解的存在性

对于

$X_{m, M}(0, T)$

，若选择M足够的小，不妨设为δ₀，利用Sobolev嵌入定理，存在m₀ > 0，使得

(15)

$\begin{align*}& 0<\frac{3}{4} v_{-} \leqslant V+\phi_{y} \leqslant \frac{5}{4} v_{-} \\& \inf\limits_{y \in \mathbb{R}, t \in(0, T)} 3 \chi^{2}-1 \geqslant m_{0}>0\end{align*}$

因此，空间

$X_{m, M}(0, T)$

可以简化为如下形式。

$X_{\delta_{0}}(0, T)=\left\{(\phi, \psi, \chi) \mid(\phi, \psi) \in C\left(0, T ; H^{2}\right)\right., \\\chi^{2}-1 \in C\left(0, T ; L^{2}\right), \chi_{y} \in C\left(0, T ; H^{2}\right), \\\left.\sup\limits_{t \in(0, T)}\left(\|(\phi, \psi)\|_{2}+\left\|\chi_{\nu}\right\|_{2}+\left\|\chi^{2}(t)-1\right\|\right) \leqslant \delta_{0}\right\}$

命题2 假定

$(\phi, \psi, \chi) \in X_{\delta}([0, +\infty))$

是Cauchy问题（11）在某个T > 0时的解，那么存在与T无关的正常数δ₀和C，若满足

(16)

$\begin{equation*}\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}+\left\|\chi_{0 v}\right\|_{2}+\left\|\chi_{0}^{2}-1\right\|+\left|v_{+}-v_{-}\right| \leqslant \delta_{0} \end{equation*}$

那么

(17)

$\begin{align*}& \|(\phi, \psi)(t)\|_{2}^{2}+\left\|\chi_{y}(t)\right\|_{2}^{2}+\int_{0}^{+\infty}\left(\left\|\phi_{y}\right\|_{1}^{2}+\left\|\psi_{y}\right\|_{2}^{2}+\right. \\& \left.\left\|\chi_{y}\right\|_{3}^{2}\right) \mathrm{d} \tau+\left\|\chi^{2}(t)-1\right\|^{2} \leqslant C\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\right. \\& \left.\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{2}^{2}\right) \end{align*}$

和

(18)

$\begin{align*}& \int_{0}^{+\infty}\left|\frac{\mathrm{d}}{\mathrm{~d} t}\left(\left\|\phi_{y y}\right\|^{2}\right)\right|+\left|\frac{\mathrm{d}}{\mathrm{~d} t}\left(\left\|\psi_{y y}\right\|^{2}\right)\right|+\left|\frac{\mathrm{d}}{\mathrm{~d} t}\left(\left\|\chi_{y y}\right\|^{2}\right)\right| \mathrm{d} \tau \leqslant \\& C\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{2}^{2}\right) \end{align*}$

成立。

下面将给出命题2的证明。为了方便得到先验估计中v的高阶估计，参考文献[13]中的方法引入有效速度

$h=u-\frac{v_{y}}{v}, H(y)=U-\frac{V_{y}}{V}$

，并进一步假设有效速度的反导数为

$\tilde{\psi}=\int_{-\infty}^{y}(h(z, t)-H(z+\alpha)) \mathrm{d} z$

代入原方程可以得到以下方程。

(19)

$\left\{\begin{array}{l}\phi_{t}-s \phi_{y}-\tilde{\psi}_{y}-\frac{1}{V} \phi_{y y}=F-\frac{1}{V^{2}} V_{y} \phi_{y} \\\tilde{\psi}_{t}-s \tilde{\psi}_{y}+p^{\prime}(V) \phi_{y}=G-\frac{1}{2} \frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}} \\\chi_{t}-s \chi_{y}=-\left(V+\phi_{y}\right) \mu \\\mu=\left(\chi^{3}-\chi\right)-\left(\frac{\chi_{y}}{V+\phi_{y}}\right)_{y}\end{array}\right.$

初值满足

$\begin{aligned}& \left.(\phi, \tilde{\psi}, \chi)\right|_{t=0}=\left(\phi_{0}, \tilde{\psi}_{0}, \chi_{0}\right)(y) \xrightarrow{y \rightarrow+\infty}(0, 0, \pm 1) \\& \phi_{0}, \tilde{\psi}_{0} \in L^{2}(\mathbb{R}), \left(v_{0}, h_{0}\right)-(V, H) \in H^{2}(\mathbb{R}) \cap L^{1}(\mathbb{R})\end{aligned}$

其中

$\begin{aligned}& F=\left(\frac{1}{v}-\frac{1}{V}\right) \phi_{y y}+\left(\frac{1}{v}-\frac{1}{V}+\frac{1}{V^{2}} \phi_{y}\right) V_{y} \\& G=-\left(p(v)-p(V)-p^{\prime}(V)(v-V)\right)\end{aligned}$

引理1 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，以下不等式成立。

(20)

$|\chi(x, t)| \leqslant 1, \forall(x, t) \in(-\infty, +\infty) \times[0, T]$

并且满足估计

(21)

$\begin{align*}& \left\|\left(\phi, \tilde{\psi}, \chi^{2}-1\right)(t)\right\|^{2}+\int_{0}^{t}\left(\left\|\phi_{y}\right\|^{2}+\left\|\sqrt{V_{y}} \tilde{\psi}\right\|^{2}+\right. \\& \left.\left\|\chi^{3}-\chi\right\|^{2}\right) \mathrm{d} \tau+\int_{0}^{t}\left(\left\|\chi_{y}\right\|^{2}\right) \mathrm{d} \tau \leqslant \\& C\left\|\left(\phi_{0}, \tilde{\psi}_{0}, \chi_{0}^{2}-1\right)\right\|^{2}+C \int_{0}^{t}\left\|\phi_{y y}\right\|^{2} \mathrm{~d} \tau \end{align*}$

证明考虑式（1）中第三、四式，利用抛物方程极大值原理，可得式（20）。将式（19）第一式乘以ϕ，式（19）第二式乘以

$-\frac{\tilde{\psi}}{p^{\prime}(V)}$

，式（19）第三式乘以

$\chi^{3}-\chi$

，各式相加并将结果关于y和t在(-∞, +∞)×[0, t)上积分得

$\begin{aligned}& \left.\int_{-\infty}^{+\infty}\left(\frac{\phi^{2}}{2}-\frac{\tilde{\psi}^{2}}{2 p^{\prime}(V)}+\frac{\left(\chi^{2}-1\right)^{2}}{4}\right) \mathrm{d} y\right|_{0} ^{t}+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(3 \chi^{2}-1\right) \chi_{y}^{2} \mathrm{~d} y \mathrm{~d} \tau+\\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(\left(V+\phi_{y}\right)\left(\chi^{3}-\chi\right)^{2}+\frac{1}{V} \phi_{y}^{2}+\right. \\& \left.\frac{s}{2} \frac{p^{\prime \prime}(V)}{\left(p^{\prime}(V)\right)^{2}} V_{y} \tilde{\psi}^{2}\right) \mathrm{d} y \mathrm{~d} \tau= \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(\frac{\chi_{y}^{2}}{2\left(V+\phi_{y}\right)^{2}} \frac{\tilde{\psi}}{p^{\prime}(V)}-\right. \\& \left.\frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi\left(\chi^{2}-1\right)\right) \mathrm{d} y \mathrm{~d} \tau+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(F \phi-\frac{G \tilde{\psi}}{p^{\prime}(V)}\right) \mathrm{d} y \mathrm{~d} \tau\end{aligned}$

利用式（7）可得

(22)

$\begin{align*}& |F|=\left|\left(\frac{1}{v}-\frac{1}{V}\right) \phi_{y y}+\left(\frac{1}{v}-\frac{1}{V}+\frac{1}{V^{2}} \phi_{y}\right) V_{y}\right| \leqslant \\& C\left(\left|\phi_{y}\right|\left|\phi_{y y}\right|+\delta_{1}^{2} \mathrm{e}^{-\delta_{i, l}, y \mid}\left|\phi_{y}\right|^{2}\right)\end{align*}$

(23)

$|G| \leqslant C \phi_{y}^{2}$

结合式（15）和式（12），得到

(24)

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty} F \phi \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant C \delta \int_{0}^{t}\left(\left\|\phi_{y}\right\|^{2}+\left\|\phi_{y y}\right\|^{2}\right) \mathrm{d} \tau $

(25)

$\begin{equation*}\left|\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{G \tilde{\psi}}{p^{\prime}(V)} \mathrm{d} y \mathrm{~d} \tau\right| \leqslant C \delta \int_{0}^{t}\left\|\phi_{y}\right\|^{2} \mathrm{~d} \tau \end{equation*}$

再次利用式（15）、（12），可得

(26)

$\begin{equation*}\left|\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{\chi_{y}^{2}}{2\left(V+\phi_{y}\right)^{2}} \frac{\tilde{\psi}}{p^{\prime}(V)} \mathrm{d} y \mathrm{~d} \tau\right| \leqslant C \delta \int_{0}^{t}\left\|\chi_{y}\right\|^{2} \mathrm{~d} \tau \end{equation*}$

和

(27)

$\begin{align*}& \left|\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi\left(\chi^{2}-1\right) \mathrm{d} y \mathrm{~d} \tau\right| \leqslant \\& \frac{\delta}{2} \int_{0}^{t}\left\|\chi_{y}\right\|^{2} \mathrm{~d} \tau+C \delta_{1} \int_{0}^{t}\left\|\chi^{3}-\chi\right\|^{2} \mathrm{~d} \tau+ \\& C \int_{0}^{t}\left\|\phi_{y y}\right\|^{2} \mathrm{~d} \tau \end{align*}$

结合式（24）~（27）和式（6）、（7），选取δ、δ₁足够的小，可得式（21），引理1证毕。

引理2 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，则以下不等式成立。

(28)

$\begin{align*}& \left\|\left(\phi_{y}, \tilde{\psi}_{y}, \chi_{y}\right)(t)\right\|^{2}+ \\& \int_{0}^{t}\left(\left\|\phi_{y y}\right\|^{2}+\left\|\tilde{\psi}_{y}\right\|^{2}+\left\|\chi_{y y}\right\|^{2}\right) \mathrm{d} \tau \leqslant \\& C\left\|\left(\phi_{0 y}, \tilde{\psi}_{0 y}, \chi_{0 y}\right)\right\|^{2} \end{align*}$

证明将式（19）第一式乘以-ϕ_yy，式（19）第二式乘以

$-\frac{\tilde{\psi}_{y y}}{p^{\prime}(V)}$

，式（19）第三式乘以χ_yy，相加并将结果关于y和t在(-∞, +∞)×[0, t)上积分得

(29)

$\left.\int_{-x}^{+\infty}\left(\frac{\phi_{y}^{2}}{2}-\frac{\tilde{\psi}_{y}^{2}}{2 p^{\prime}(V)}+\chi_{y}^{2}\right) \mathrm{d} y\right|_{0} ^{t}+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(V+\phi_{y}\right)\left(3 \chi^{2}-1\right) \chi_{y}^{2} \mathrm{~d} y \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(\chi_{y y}^{2}+\frac{\phi_{y y}^{2}}{V}+\frac{s p^{\prime \prime}(V) V_{y}}{\left(p^{\prime}(V)\right)^{2}} \tilde{\psi}_{y}^{2}-F \phi_{y y}+\right.\\\left.\frac{V_{y}}{V^{2}} \phi_{y} \phi_{y y}\right) \mathrm{d} y \mathrm{~d} \tau=\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(-\left(\frac{1}{p^{\prime}(V)}\right)_{y} p^{\prime}(V) \phi_{y} \tilde{\psi}_{y}-\frac{1}{p^{\prime}(V)} G_{y} \tilde{\psi}_{y}\right) \mathrm{d} y \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(-\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right) \chi_{y}+\frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi_{y y}\right) \mathrm{dyd} \tau$

利用式（12）、（22）和（23），得到

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}\left(F \phi_{y y}-\frac{V_{y}}{V^{2}} \phi_{y} \phi_{y y}\right) \mathrm{d} y \mathrm{~d} \tau\right| \leqslant C \int_{0}^{t} \delta\left\|\phi_{y y}\right\|^{2}+\\\left\|\phi_{y}\right\|^{2} \mathrm{~d} \tau$

以及

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}\left[-\left(\frac{1}{p^{\prime}(V)}\right)_{y} p^{\prime}(V) \phi_{y} \tilde{\psi}_{y}-\frac{1}{p^{\prime}(V)} G_{y} \tilde{\psi}_{y}\right] \mathrm{d} y \mathrm{~d} \tau\right| \leqslant\\\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{s}{2} \frac{p^{\prime \prime}(V) V_{y}}{\left(p^{\prime}(V)\right)^{2}} \tilde{\psi}_{y}^{2} \mathrm{~d} y \mathrm{~d} \tau+C \delta \int_{0}^{t}\left(\left\|\tilde{\psi}_{y}\right\|^{2}+\left\|\tilde{\psi}_{y y}\right\|^{2}\right) \mathrm{d} \tau$

更进一步得到

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}\left(\frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi_{y y}-\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right) \chi_{y}\right) \mathrm{d} y \mathrm{~d} \tau\right| \leqslant\\\frac{1}{2} \int_{0}^{t}\left(\left\|\chi_{y y}\right\|^{2}+\frac{1}{v_{+}}\left\|\phi_{y y}\right\|^{2}\right) \mathrm{d} \tau+C\left(\delta+\left|v_{+}-v_{-}\right|\right) \int_{0}^{t}\left\|\chi^{3}-\chi\right\|^{2} \mathrm{~d} \tau$

利用上述估计结合引理1，选取δ、δ₁足够的小，得到

$\begin{aligned}& \int_{-\infty}^{+\infty}\left(\phi_{y}^{2}+\tilde{\psi}_{y}^{2}+\chi_{y}^{2}\right) \mathrm{d} y+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(\phi_{y y}^{2}+V_{y} \tilde{\psi}_{y}^{2}+\chi_{y y}^{2}+\chi_{y}^{2}\right) \mathrm{d} y \mathrm{~d} \tau \leqslant \\& C \int_{-\infty}^{+\infty}\left(\phi_{0 y}^{2}+\tilde{\psi}_{0 y}^{2}+\chi_{0 y}^{2}\right) \mathrm{d} y+C \int_{0}^{t} \int_{-\infty}^{+\infty} \phi_{y}^{2} \mathrm{~d} y \mathrm{~d} \tau\end{aligned}$

再利用引理1，得到

(30)

$\begin{align*}& \left\|\left(\phi_{y}, \tilde{\psi}_{y}, \chi_{y}\right)(t)\right\|^{2}+ \\& \int_{0}^{t}\left(\left\|\phi_{y y}\right\|^{2}+\left\|\sqrt{V_{y}} \tilde{\psi}_{y}\right\|^{2}+\left\|\chi_{y y}\right\|^{2}\right) \mathrm{d} \tau \leqslant \\& C\left\|\left(\phi_{0 y}, \tilde{\psi}_{0 y}, \chi_{0 y}\right)\right\|^{2} \end{align*}$

将式（19）第一式乘以

$\tilde{\psi}_{y}$

，式（19）第二式关于y求导并乘以ϕ，两式相加并将结果关于y和t在(-∞, +∞)×[0, t)上积分，再利用式（29）、（30）和引理1，引理2得证。

结合引理1和引理2可以得到

(31)

$\|(\phi, \tilde{\psi})(t)\|_{1}^{2}+\left\|\chi^{2}-1\right\|^{2}+\left\|\chi_{y}\right\|^{2}+\int_{0}^{t}\left\|\phi_{y}\right\|_{1}^{2} \mathrm{~d} \tau+\\\int_{0}^{t}\left(\left\|\sqrt{V_{y}} \tilde{\psi}\right\|^{2}+\left\|\tilde{\psi}_{y}\right\|^{2}+\left\|\chi_{y}\right\|_{1}^{2}+\left\|\chi^{3}-\chi\right\|^{2}\right) \mathrm{d} \tau \leqslant\\C\left(\left\|\left(\phi_{0}, \tilde{\psi}_{0}\right)\right\|_{1}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|^{2}\right)$

引理3 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，则以下不等式成立。

(32)

$\left(\|\phi\|_{1}^{2}+\|\psi\|^{2}\right)(t)+\left\|\chi^{2}-1\right\|^{2}+\left\|\chi_{y}\right\|^{2}+\\\int_{0}^{t}\left(\left\|\phi_{y}\right\|_{1}^{2}+\left\|\psi_{y}\right\|^{2}+\left\|\chi_{y}\right\|_{1}^{2}+\left\|\chi^{3}-\chi\right\|^{2}\right) \mathrm{d} \tau \leqslant\\C\left(\left\|\psi_{0}\right\|_{1}^{2}+\left\|\phi_{0}\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|^{2}\right)$

证明利用有效速度的定义

$\begin{aligned}& \tilde{\psi}(y, t)=\int_{-\infty}^{y}[u(\xi, t)-U(\xi, t)] \mathrm{d} y- \\& {[g(v(\xi, t))-g(V(\xi, t))]}\end{aligned}$

其中

$g(v)=\frac{v_{y}}{v}$

，由于

$\tilde{\psi}(y, t) \leqslant \psi(y, t)+C\left|\phi_{y}(y, t)\right|$

，可以推出

$|\tilde{\psi}(y, t)| \leqslant|\psi(y, t)|+C\left|\phi_{y}(y, t)\right|$

。类似地得到

$\|\psi\|^{2}(t) \leqslant\|\tilde{\psi}\|^{2}(t)+C\left\|\phi_{y}\right\|^{2}(t)$

。利用低阶估计式（31），得到

$\|\psi\|^{2}(t) \leqslant\|\tilde{\psi}\|^{2}(t)+C\|\phi\|_{1}^{2}(t) \leqslant C\left\|\left(\phi_{0}, \tilde{\psi}_{0}\right)\right\|_{1}^{2}$

类似地可以得到

$\int_{0}^{t}\left\|\psi_{y}\right\|^{2} \mathrm{~d} \tau \leqslant \int_{0}^{t}\left(\left\|\tilde{\psi}_{y}\right\|^{2}+C\left\|\phi_{y}\right\|_{1}^{2}\right) \mathrm{d} \tau \leqslant\\C\left\|\left(\phi_{0}, \tilde{\psi}_{0}\right)\right\|_{1}^{2}$

以及

$\left\|\tilde{\psi}_{0}\right\|_{1}^{2} \leqslant\left\|\psi_{0}\right\|_{1}^{2}+C\left\|\phi_{0}\right\|_{2}^{2}$

结合以上表达式，得到引理3的证明。

引理4 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，则以下不等式成立。

(33)

$\begin{gather*}\left\|\left(\chi_{y}, \psi_{y}\right)(t)\right\|^{2}+\int_{0}^{t}\left\|\psi_{y y}\right\|^{2} \mathrm{~d} \tau+\int_{0}^{t}\left\|\chi_{y}\right\|_{1}^{2} \mathrm{~d} \tau \leqslant \\C\left(\left\|\psi_{0}\right\|_{1}^{2}+\left\|\phi_{0}\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|^{2}\right) \end{gather*}$

证明将式（11）第二式乘以-ψ_yy，式（11）第三式乘以χ_yy，两者相加并将结果关于t和y在(-∞, +∞)×[0, t)上积分，得到

$\begin{aligned}& \frac{1}{2}\left\|\left(\chi_{y}, \psi_{y}\right)(t)\right\|^{2}+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(\left(V+\phi_{y}\right)\left(3 \chi^{2}-1\right) \chi_{y}^{2}+W \psi_{y y}\right) \mathrm{d} y \mathrm{~d} \tau+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left(\chi_{y y}^{2}+\frac{\psi_{y y}^{2}}{V}\right) \mathrm{d} y \mathrm{~d} \tau+ \\& \int_{0}^{t} \int_{-\infty}^{+\infty}\left[\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right) \chi_{y}\right] \mathrm{d} y \mathrm{~d} \tau=\end{aligned}$

$\begin{aligned}\frac{1}{2}\left\|\left(\chi_{0 y}, \psi_{0 y}\right)(t)\right\|^{2}+\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{1}{2} \psi_{y y} \frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}} \mathrm{~d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{+\infty} f\left(V, U_{y}\right) \phi_{y} \psi_{y y} \mathrm{~d} y \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(\frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi_{y y}\right) \mathrm{d} y \mathrm{~d} \tau\end{aligned}$

利用Cauchy不等式，得到

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}-f\left(V, U_{y}\right) \phi_{y} \psi_{y y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant \varepsilon \int_{0}^{t}\left\|\psi_{y y}\right\|^{2} \mathrm{~d} \tau+\\C_{\varepsilon} \int_{0}^{t}\left\|\phi_{y}\right\|^{2} \mathrm{~d} \tau$

利用Sobolev不等式，得到

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}-W \psi_{y y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant C \delta \int_{0}^{t}\left(\left\|\phi_{y}\right\|^{2}+\left\|\psi_{y y}\right\|^{2}\right) \mathrm{d} \tau\\\left|\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{1}{2} \psi_{y y} \frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant C \int_{0}^{t}\left\|\chi_{y}\right\|^{2} \mathrm{~d} \tau$

易知

$\left|\int_{0}^{t} \int_{-\infty}^{+\infty}\left[-\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right) \chi_{y}+\frac{V_{y}+\phi_{y y}}{V+\phi_{y}} \chi_{y} \chi_{y y}\right] \mathrm{d} y \mathrm{~d} \tau\right| \leqslant\\\frac{1}{2} \int_{0}^{t}\left(\left\|\chi_{y y}\right\|^{2}+\frac{1}{v_{+}}\left\|\phi_{y y}\right\|^{2}\right) \mathrm{d} \tau+C\left(\delta+\left|v_{+}-v_{-}\right|\right) \int_{0}^{t}\left\|\chi^{3}-\chi\right\|^{2} \mathrm{~d} \tau$

综上所述引理4得证。

引理5 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，则以下不等式成立。

(34)

$\left\|\chi_{y y}\right\|^{2}+\left\|\phi_{y y}\right\|^{2}+\int_{0}^{t}\left(\left\|\phi_{y y}\right\|^{2}+\left\|\chi_{y y y}\right\|^{2}\right) \mathrm{d} \tau \leqslant\\C\left(\left\|\phi_{0}\right\|_{2}^{2}+\left\|\psi_{0}\right\|_{1}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{1}^{2}\right)+\\C \delta \int_{0}^{t}\left\|\psi_{y y y}\right\|^{2} \mathrm{~d} \tau$

证明将式（11）第一式关于y微分，利用式（11）第二式得到

(35)

$\frac{\phi_{t y}}{V}-\frac{s \phi_{y y}}{V}+f\left(V, U_{y}\right) \phi_{y}=\psi_{t}-s \psi_{y}-W+\\\frac{1}{2} \cdot \frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}}$

将式（11）第三式对y求导两次并乘以χ_yy，将式（35）关于y微分并乘以ϕ_yy，两者相加并将结果关于t和y在(-∞, +∞)×[0, t)上积分，得到

$\left.\frac{1}{2} \int_{-\infty}^{+\infty}\left(\chi_{y y}^{2}+\frac{\phi_{y y}^{2}}{V}\right)\right|_{0} ^{t} \mathrm{~d} y+\int_{0}^{t} \int_{-\infty}^{+\infty} \chi_{y y y}^{2} \mathrm{~d} y \mathrm{~d} \tau+ \\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(f\left(V, U_{y}\right)+\frac{s}{2}\left(\frac{1}{V}\right)_{y}\right) \phi_{y y}^{2} \mathrm{~d} y \mathrm{~d} \tau=$

$\left.\int_{-\infty}^{+\infty} \psi_{y} \phi_{y y}\right|_{0} ^{t} \mathrm{~d} y+\int_{0}^{t}\left\|\psi_{y y}\right\|^{2} \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{\infty} \frac{1}{2} \phi_{y y}\left(\frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}}\right) \mathrm{d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{\infty}\left(\frac{1}{V}\right)_{y} \psi_{y y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{\infty} f\left(V, U_{y}\right)_{y} \phi_{y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right)+\left(V+\phi_{y}\right)\left(3 \chi^{2}-1\right) \chi_{y}\right) \chi_{y y} \mathrm{~d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(V_{y}+\phi_{y y}\right)\left(\frac{\chi_{y y} \chi_{y y}}{V+\phi_{y}}+\left(\frac{1}{V+\phi_{y}}\right)_{y} \chi_{y} \chi_{y y y}\right) \mathrm{d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(\left(V+\phi_{y}\right)\left(\frac{1}{V+\phi_{y}}\right)_{y y} \chi_{y} \chi_{y y y}\right) \mathrm{d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{\infty} W_{y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(2\left(V+\phi_{y}\right)\left(\frac{1}{V+\phi_{y}}\right)_{y} \chi_{y y} \chi_{y y y}\right) \mathrm{d} y \mathrm{~d} \tau$

利用Cauchy不等式，得到

$\int_{-\infty}^{\infty} \psi_{y} \phi_{y y} \mathrm{~d} y \leqslant \varepsilon\left\|\phi_{y y}\right\|^{2}+C_{\varepsilon}\left\|\psi_{y}\right\|^{2}$

而

$\int_{0}^{t}\left\|\psi_{y y}\right\|^{2} \mathrm{~d} \tau$

由引理4控制，利用Cauchy不等式，可得

$\begin{aligned}& \left|-\int_{0}^{t} \int_{-\infty}^{\infty} W_{y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant \varepsilon \int_{0}^{t}\left\|\phi_{y y}\right\|^{2} \mathrm{~d} \tau+ \\& C_{\varepsilon} \delta \int_{0}^{t}\left(\left\|\phi_{y}\right\|_{1}^{2}+\left\|\psi_{y}\right\|_{2}^{2}\right) \mathrm{d} \tau \\ & \left|\int_{0}^{t} \int_{-\infty}^{\infty}-\left(\frac{1}{V}\right)_{y} \psi_{y y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant \varepsilon \int_{0}^{t}\left\|\phi_{y y}\right\|^{2} \mathrm{~d} \tau+\\ & C_{\varepsilon} \int_{0}^{t}\left\|\psi_{y y}\right\|^{2} \mathrm{~d} \tau\end{aligned}$

利用f(V, U_y)_y < C，可得

$\left|-\int_{0}^{t} \int_{-\infty}^{\infty} f\left(V, U_{y}\right)_{y} \phi_{y} \phi_{y y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant \varepsilon \int_{0}^{t}\left\|\phi_{y y}\right\|^{2} \mathrm{~d} \tau+\\C_{\varepsilon} \int_{0}^{t}\left\|\phi_{y}\right\|^{2} \mathrm{~d} \tau\\\left|\int_{0}^{t} \int_{-\infty}^{+\infty} \frac{1}{2} \phi_{y y}\left(\frac{\chi_{y}^{2}}{\left(V+\phi_{y}\right)^{2}}\right)_{y} \mathrm{~d} y \mathrm{~d} \tau\right| \leqslant\\C \int_{0}^{t}\left(\left\|\chi_{y}\right\|^{2}+\left\|\chi_{y y}\right\|^{2}\right) \mathrm{d} \tau\\\mid \int_{0}^{t} \int_{-\infty}^{+\infty}\left[\left(V_{y}+\phi_{y y}\right)\left(\chi^{3}-\chi\right) \chi_{y y y}\right] \mathrm{d} y \mathrm{~d} \tau+\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left[\left(V+\phi_{y}\right)\left(3 \chi^{2}-1\right) \chi_{y} \chi_{y y y}\right] \mathrm{d} y \mathrm{~d} \tau \mid \leqslant\\C \int_{0}^{t}\left(\left\|\chi_{y}\right\|^{2}+\left\|\chi^{3}-\chi\right\|^{2}\right) \mathrm{d} \tau+\frac{1}{8} \int_{0}^{t}\left\|\chi_{y y y}\right\|^{2} \mathrm{~d} \tau\\\left\lvert\, -\int_{0}^{t} \int_{-\infty}^{+\infty}\left(V_{y}+\phi_{y y}\right)\left[\left(\frac{1}{V+\phi_{y}}\right)_{y} \chi_{y} \chi_{y y}\right] \mathrm{d} y \mathrm{~d} \tau-\right.\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left(V_{y}+\phi_{y y}\right)\left(\frac{\chi_{y y} \chi_{y y}}{V+\phi_{y}}\right) \mathrm{d} y \mathrm{~d} \tau-\\\int_{0}^{t} \int_{-\infty}^{+\infty}\left[\left(V+\phi_{y}\right)\left(\frac{1}{V+\phi_{y}}\right)_{y y} \chi_{y} \chi_{y y y}\right] \mathrm{d} y \mathrm{~d} \tau-\\\left.\int_{0}^{t} \int_{-\infty}^{+\infty}\left[2\left(V+\phi_{y}\right)\left(\frac{1}{V+\phi_{y}}\right)_{y} \chi_{y y} \chi_{y y y}\right] \mathrm{d} y \mathrm{~d} \tau \right\rvert\, \leqslant\\C \int_{0}^{t}\left(\left\|\chi_{y}\right\|^{2}+\left\|\chi_{y y}\right\|^{2}\right) \mathrm{d} \tau+\frac{1}{8} \int_{0}^{t}\left\|\chi_{y y y}\right\|^{2} \mathrm{~d} \tau$

再利用引理3和引理4，引理5得证。

引理6 假定(ϕ, ψ, χ)∈X_δ(0, T)是式（11）的解，那么对于t∈[0, T]，以下不等式成立。

(36)

$\left\|\chi_{y y y}\right\|^{2}+\left\|\psi_{y y}\right\|^{2}+\int_{0}^{t}\left\|\psi_{y y y}\right\|^{2} \mathrm{~d} \tau+\int_{0}^{t}\left\|\chi_{y y y}\right\|^{2} \mathrm{~d} \tau \leqslant\\C\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{2}^{2}\right)$

证明运用与引理5类似的方法将得到三阶导数的估计，此处省去细节。

下面证明命题2中的式（18）。将式（11）第一式关于y微分两次，并将结果乘以ϕ_yy，然后关于y在R上积分，得到

(37)

$\frac{1}{2} \frac{\mathrm{~d}}{\mathrm{~d} t}\left(\left\|\phi_{y y}\right\|^{2}\right)=-\int_{-\infty}^{+\infty}\left(\psi_{y y} \phi_{y y y}\right) \mathrm{d} y$

利用式（17），得到

(38)

$\int_{0}^{+\infty}\left|\frac{\mathrm{d}}{\mathrm{d} t}\left(\left\|\phi_{y y}\right\|^{2}\right)\right| \mathrm{d} \tau \leqslant C_{0}\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\right.\\\left.\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{2}^{2}\right)$

类似地，可以得到

(39)

$\int_{0}^{+\infty}\left[\left|\frac{\mathrm{d}}{\mathrm{d} t}\left(\left\|\psi_{y y}\right\|^{2}\right)\right|+\left|\frac{\mathrm{d}}{\mathrm{d} t}\left(\left\|\chi_{y}\right\|^{2}\right)\right|\right] \mathrm{d} \tau \leqslant \\ C_{0}\left(\left\|\left(\phi_{0}, \psi_{0}\right)\right\|_{2}^{2}+\left\|\chi_{0}^{2}-1\right\|^{2}+\left\|\chi_{0 y}\right\|_{2}^{2}\right)$

由式（37）~（39）可以推出先验估计式（18），命题2证毕。

再利用Sobolev不等式，可以得到

$\lim\limits_{t \rightarrow 0}\left\|\phi_{y}, \psi_{y}, \chi^{2}-1\right\|_{L^{*}(\mathbb{R})}=0$

因此定理1证毕。

3 结束语

本文研究了一维等熵Navier-Stokes/Allen-Cahn方程组黏性激波解的渐进稳定性。在初值小扰动的条件下，考虑在相分离状态附近的情形，并通过引入有效速度降低了初始条件的正则性，证明了NSAC方程激波解的存在性和大时间行为。

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