Nonlinear stability of entropy waves for the Euler equations

Abstract

In this article, we consider a class of the contact discontinuity for the full compressible Euler equations, namely the entropy wave, where the velocity is continuous across the interface while the density and the entropy can have jumps. The nonlinear stability of entropy waves is a longstanding open problem in multi-dimensional hyperbolic conservation laws. The rigorous treatments are challenging due to the characteristic discontinuity nature of the problem (G.-Q. Chen and Y.-G. Wang in Nonlinear partial differential equations, Volume 7 of Abel Symp.(2012)). In this article, we discover that the Taylor sign condition plays an essential role in the nonlinear stability of entropy waves. By deriving the evolution equation of the interface in the Eulerian coordinates, we relate the Taylor sign condition to the hyperbolicity of this evolution equation, which reveals a stability condition of the entropy wave. With the optimal regularity estimates of the interface, we can derive the a priori estimates without loss of regularity.

Appendices

Elliptic estimates

For the one-phase elliptic system

$$\begin{aligned} {\left\{ \begin{array}{ll} \nabla \times u=\omega , \quad \nabla \cdot u=\sigma , &{} \text {in } \Omega _f^-, \\ u\cdot N=\theta , &{} \text {on } \Gamma _f, \\ u\cdot n_-=0, \quad \int _{{\mathbb {T}}^2}u_j\textrm{d}{\overline{x}}=\alpha _j \,(j=1,\,2), &{} \text {on } \Gamma ^-, \\ \end{array}\right. } \end{aligned}$$

(A.1)

we have the following existence result given by Proposition 5.1 in [34] (see also [8, 33]):

Lemma A.1

Assume that $f\in H^{\kappa -\frac{1}{2}}({\mathbb {T}}^2)$ with $\kappa >\frac{5}{2}$. For $s\in [2,\,\kappa ]$, let $(\omega ,\,\sigma )\in H^{s-2}(\Omega _f^-)$ and $\theta \in H^{s-\frac{3}{2}}({\mathbb {T}}^2)$ be such that

$$\begin{aligned}{} & {} \int _{\Omega _f^-}\sigma \textrm{d}x =\int _{{\mathbb {T}}^2}\theta \textrm{d}{\overline{x}},\\{} & {} \nabla \cdot \omega =0, \,\text {in } \Omega _f^-, \quad \int _{\Gamma ^-}\omega _3\textrm{d}{\overline{x}}=0. \end{aligned}$$

Then there exists a unique $u\in H^{s-1}(\Omega _f^-)$ to the system (A.1) such that

$$\begin{aligned} \big \Vert u\big \Vert _{H^{s-1}(\Omega _f^-)}\le & {} C(\big \Vert f\big \Vert _{H^{\kappa -\frac{1}{2}}}) \Big \{\big \Vert (\omega ,\,\sigma )\big \Vert _{H^{s-2}(\Omega _f^-)} +\big \Vert \theta \big \Vert _{H^{s-\frac{3}{2}}(\Gamma _f)} \nonumber \\{} & {} +|\alpha _1|+|\alpha _2|\Big \}. \end{aligned}$$

(A.2)

The regularity of the solution of the one-phase elliptic system (A.1) was improved in [8] (see also [39]) by using tangential derivatives for the boundary condition on the surface $\Gamma _f$:

Lemma A.2

Assume that $f\in H^{\kappa -\frac{1}{2}}({\mathbb {T}}^2)$ with $\kappa >\frac{5}{2}$. For $s\in [2,\,\kappa ]$, there holds that

$$\begin{aligned} \begin{aligned} \big \Vert u\big \Vert _{H^{s}(\Omega _f^-)} \le&C(\big \Vert f\big \Vert _{H^{\kappa -\frac{1}{2}}})\Big \{\big \Vert \nabla \times u\big \Vert _{H^{s-1}(\Omega _f^-)} +\big \Vert \nabla \cdot u\big \Vert _{H^{s-1}(\Omega _f^-)} \\&\quad \quad +\sum _{i=1,2}\big \Vert {\overline{\partial }}_i u\cdot N \big \Vert _{H^{s-\frac{3}{2}}(\Gamma _f)} +\big \Vert u\big \Vert _{L^2(\Omega _f^-)}\Big \}. \end{aligned} \end{aligned}$$

(A.3)

Clearly, these two results also hold for the one-phase elliptic systems in $\Omega ^+_f$ in a similar fashion.

Paradifferential operators and commutator estimates

In this appendix, we shall recall some basic facts on paradifferential operators from [26].

We first introduce the symbols with limited spatial smoothness. Let $W^{k,\infty }({\mathbb {R}}^d)$ be the usual Sobolev spaces for $k\in {\mathbb {N}}$.

Definition B.1

Given $\mu \in [0,\,1]$ and $m\in {\mathbb {R}}$, we denote by $\Gamma ^m_{\mu }({\mathbb {R}}^d)$ the space of locally bounded functions $a(x,\,\xi )$ on ${\mathbb {R}}^d\times {\mathbb {R}}^d\backslash \{0\}$, which are $C^{\infty }$ with respect to $\xi$ for $\xi \ne 0$ such that, for all $\alpha \in {\mathbb {N}}^d$ and $\xi \ne 0$, the function $x\rightarrow \partial _{\xi }^{\alpha }a(x,\,\xi )$ belongs to $W^{\mu ,\infty }$ and there exists a constant $C_{\alpha }$ such that

$$\begin{aligned} \big \Vert \partial _{\xi }^{\alpha }a(\cdot ,\,\xi )\big \Vert _{W^{\mu ,\infty }} \le C_{\alpha }(1+|\xi |)^{m-|\alpha |}, \quad \quad \forall \,|\xi |\ge \frac{1}{2}. \end{aligned}$$

The seminorm of the symbol is defined as

$$\begin{aligned} M_{\mu }^m(a):=\sup _{|\alpha |\le \frac{3d}{2}+1+\mu } \sup _{|\xi |\ge \frac{1}{2}} \big \Vert (1+|\xi |)^{-m+|\alpha |}\partial _{\xi }^{\alpha }a(\cdot ,\,\xi )\big \Vert _{W^{\mu ,\infty }} \end{aligned}$$

If a is a function independent of $\xi$, then

$$\begin{aligned} M^0_{\mu }(a)=\big \Vert a\big \Vert _{W^{\mu ,\infty }}. \end{aligned}$$

Definition B.2

Given a symbol a, the paradifferential operator $T_a$ is defined by

$$\begin{aligned} \widehat{T_au}(\xi ):=(2\pi )^{-d}\int _{{\mathbb {R}}^d} \chi (\xi -\eta ,\,\eta ) \widehat{a}(\xi -\eta ,\,\eta ) \psi (\eta )\widehat{u}(\eta )\textrm{d}\eta , \end{aligned}$$

(B.1)

where $\widehat{a}$ is the Fourier transform of a with respect to the first variable. $\chi (\xi ,\,\eta )\in C^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d)$ is an admissible cutoff function, that is, there exist $0<\varepsilon _1<\varepsilon _2$ such that

$$\begin{aligned} \chi (\xi ,\,\eta )=1 \quad \text {for } |\xi |\le \varepsilon _1|\eta |, \quad \quad \chi (\xi ,\,\eta )=0 \quad \text {for } |\xi |\ge \varepsilon _2|\eta |, \end{aligned}$$

and

$$\begin{aligned} |\partial _{\xi }^{\alpha }\partial _{\eta }^{\beta }\chi (\xi ,\,\eta )| \le C_{\alpha ,\beta } (1+|\eta |)^{-|\alpha |-|\beta |} \quad \text {for } (\xi ,\,\eta )\in {\mathbb {R}}^d\times {\mathbb {R}}^d. \end{aligned}$$

The cutoff function $\psi (\eta )\in C^{\infty }({\mathbb {R}}^d)$ satisfies

$$\begin{aligned} \psi (\eta )=0 \quad \text {for } |\eta |\le 1, \quad \quad \psi (\eta )=1 \quad \text {for } |\eta |\ge 2. \end{aligned}$$

The admissible cutoff function $\chi (\xi ,\,\eta )$ can be chosen as

$$\begin{aligned} \chi (\xi ,\,\eta )=\sum _{k=0}^{\infty } \zeta _{k-3}(\xi )\varphi (\eta ), \end{aligned}$$

where $\zeta (\xi )=1$ for $|\xi |\le 1.1$, $\zeta (\xi )=0$ for $|\xi |\ge 1.9$, and

$$\begin{aligned} {\left\{ \begin{array}{ll} \zeta _k(\xi )=\zeta (2^{-k}\xi ) &{} \text {for } k\in {\mathbb {Z}}, \\ \varphi _0=\zeta , \quad \varphi _k=\zeta _k-\zeta _{k-1} &{} \text {for } k\ge 1. \end{array}\right. } \end{aligned}$$

We also introduce the Littlewood-Paley operators $\Delta _k,\,S_k$ defined as

$$\begin{aligned} \Delta _k u= & {} {\mathcal {F}}^{-1}(\varphi _k\widehat{u}) \quad \text {for } k\ge 0, \quad \quad \Delta _k u=0 \quad \text {for } k<0,\\ S_ku= & {} \sum _{l\le k}\Delta _lu \quad \text {for } k\in {\mathbb {Z}}. \end{aligned}$$

When the symbol a depends only on the first variable x in $T_au$, we take $\psi =1$ in (B.1). Then $T_au$ is just usual Bony’s paraproduct defined as

$$\begin{aligned} T_au=\sum _{k=0}S_{k-3}a \Delta _ku. \end{aligned}$$

(B.2)

We have the following Bony’s paraproduct decomposition:

$$\begin{aligned} au=T_au+T_ua+R(a,\,u), \end{aligned}$$

(B.3)

where the remainder term $R(a,\,u)$ is

$$\begin{aligned} R(a,\,u)=\sum _{|k-l|\le 2}\Delta _ka\Delta _lu. \end{aligned}$$

Lemma B.3

There holds that

• (1) If $s\in {\mathbb {R}}$ and $\sigma <\frac{d}{2}$, then

  $$\begin{aligned} \big \Vert T_au\big \Vert _{H^s} \lesssim \min \{ \big \Vert a\big \Vert _{L^{\infty }}\big \Vert u\big \Vert _{H^s}, \, \big \Vert a\big \Vert _{H^{\sigma }}\big \Vert u\big \Vert _{H^{s+\frac{d}{2}-\sigma }}, \, \big \Vert a\big \Vert _{H^{\frac{d}{2}}}\big \Vert u\big \Vert _{H^{s+}}\}. \end{aligned}$$

• (2) If $s>0$ and $s_1,\,s_2\in {\mathbb {R}}$ with $s_1+s_2=s+\frac{d}{2}$, then

  $$\begin{aligned} \big \Vert R(a,\,u)\big \Vert _{H^s} \lesssim \big \Vert a\big \Vert _{H^{s_1}}\big \Vert u\big \Vert _{H^{s_2}}. \end{aligned}$$

• (3) If $s>0$, $s_1\ge s$, $s_2\ge s$ and $s_1+s_2=s+\frac{d}{2}$, then

  $$\begin{aligned} \big \Vert au\big \Vert _{H^{s}} \lesssim \big \Vert a\big \Vert _{H^{s_1}}\big \Vert u\big \Vert _{H^{s_2}}. \end{aligned}$$

  (B.4)

There is also the symbolic calculus of paradifferential operator in Sobolev spaces.

Lemma B.4

Let $m,\,m'\in {\mathbb {R}}$.

• (1) If $a\in \Gamma ^m_0({\mathbb {R}}^d)$, then for any $s\in {\mathbb {R}}$,

  $$\begin{aligned} \big \Vert T_a\big \Vert _{H^s\rightarrow H^{s-m}} \lesssim M^m_0(a). \end{aligned}$$

• (2) if $a\in \Gamma ^m_{\rho }({\mathbb {R}}^d)$ and $b\in \Gamma ^{m'}_{\rho }({\mathbb {R}}^d)$ for $\rho >0$, then for any $s\in {\mathbb {R}}$,

  $$\begin{aligned} \big \Vert T_aT_b-T_{a\sharp b}\big \Vert _{H^s\rightarrow H^{s-m-m'+\rho }} \lesssim M^m_{\rho }(a)M^{m'}_{0}(b) +M^m_{0}(a)M^{m'}_{\rho }(b), \end{aligned}$$

  where

  $$\begin{aligned} a\sharp b=\sum _{|\alpha |<\rho } \partial _{\xi }^{\alpha }a(x,\,\xi ) D_x^{\alpha }b(x,\,\xi ), \quad \quad D_x=-\textrm{i}\partial _x. \end{aligned}$$

• (3) If $a\in \Gamma ^m_{\rho }({\mathbb {R}}^d)$ for $\rho \in (0,\,1]$, then for any $s\in {\mathbb {R}}$,

  $$\begin{aligned} \big \Vert (T_a)^*-T_{a^*}\big \Vert _{H^s\rightarrow H^{s-m+\rho }} \lesssim M^m_{\rho }(a), \end{aligned}$$

  where $(T_a)^*$ is the adjoint operator of $T_a$ and $a^*$ is the complex conjugate of the symbol a.

To estimate commutators, we recall a lemma from [1] (Lemma 2.15).

Lemma B.5

Consider a symbol $p=p(t,\,x,\,\xi )$ which is homogeneous of order m. There holds that

$$\begin{aligned} \big \Vert [T_p,\,\partial _t+T_u\cdot \nabla ]u\big \Vert _{H^m} \lesssim \Big \{M_0^m(p)\big \Vert u\big \Vert _{C^{1+}_{\star }} +M_0^m(D_t p)\Big \}\big \Vert u\big \Vert _{H^m}. \end{aligned}$$

(B.5)