On Non-uniqueness of Continuous Entropy Solutions to the Isentropic Compressible Euler Equations

Abstract

We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy solutions with no presence of shock. In particular, the constructed density is smooth and the momentum is $\alpha$-Hölder continuous for $\alpha <1/7$. Also, we provide a continuous entropy solution satisfying the entropy inequality strictly.

Appendix A. Some Technical Lemmas

The proof of the following two lemmas can be found in [6, Appendix]:

Lemma A.1

(Hölder norm of compositions). Suppose $F:\Omega \rightarrow \mathbb {R}$ and $\Psi : \mathbb {R}^n \rightarrow \Omega$ are smooth functions for some $\Omega \subset \mathbb {R}^m$. Then, for each $N\in \mathbb {N}$, we have

$$\begin{aligned}&\Vert \nabla ^N (F\circ \Psi )\Vert _0 \lesssim \Vert \nabla F\Vert _0 \Vert \nabla \Psi \Vert _{N-1} + \Vert \nabla F\Vert _{N-1} \Vert \Psi \Vert _0^{N-1}\Vert \Psi \Vert _N \nonumber \\&\Vert \nabla ^N (F\circ \Psi )\Vert _0 \lesssim \Vert \nabla F\Vert _0 \Vert \nabla \Psi \Vert _{N-1} + \Vert \nabla F\Vert _{N-1} \Vert \nabla \Psi \Vert _0^{N} , \end{aligned}$$

(A.1)

where the implicit constant in the inequalities depends only on n, m, and N.

Lemma A.2

Let $N\geqslant 1$. Suppose that $a\in C^\infty (\mathbb {T}^3)$ and $\xi \in C^\infty (\mathbb {T}^3;\mathbb {R}^3)$ satisfies

$$\begin{aligned} \frac{1}{C} \leqslant |\nabla \xi | \leqslant C \end{aligned}$$

for some constant $C>1$. Then, we have

$$\begin{aligned} \left| \int _{\mathbb {T}^3} a(x) e^{ik\cdot \xi } \mathrm{d}x \right| \lesssim \frac{\Vert a\Vert _N + \Vert a\Vert _0\Vert {\nabla } \xi \Vert _{N}}{|k|^N} , \end{aligned}$$

where the implicit constant in the inequality is depending on C and N, but independent of k.

The following lemmas contain various commutator estimates, used in the proof.

Lemma A.3

Let f and g be in $C^\infty ([0,T]\times \mathbb {T}^3)$ and set $f_\ell = P_{\leqslant \ell ^{-1} } f$, $g_\ell = P_{\leqslant \ell ^{-1}} g$ and $(fg)_\ell = P_{\leqslant \ell ^{-1}} (fg)$. Then, for each $N \geqslant 0$, the following holds,

$$\begin{aligned} \Vert f_\ell g_\ell - (fg)_\ell \Vert _N \lesssim _N \ell ^{2-N} \Vert f\Vert _1\Vert g\Vert _1. \end{aligned}$$

(A.2)

Proof

Since the expression that we need to estimate is localized in frequency, by Bernstein’s inequality it suffices to prove the case $N=0$. Recall now the function $\phi$ used to define the Littlewood-Paley operators and the number J, which is the maximal natural number such that $2^J \leqslant \ell ^{-1}$. Denoting by the inverse Fourier transform and by the function , a simple computation (see for instance [13]) gives

and the claim easily follows. $\square$

Lemma A.4

Let f and g be in $C^\infty ([0,T]\times \mathbb {T}^3)$ and set $f_\ell = P_{\leqslant \ell ^{-1} } f$ and $(fg)_\ell = P_{\leqslant \ell ^{-1}} (fg)$. Then, for each $N \geqslant 0$, the following holds,

$$\begin{aligned} \begin{aligned} \Vert [g, P_{\leqslant \ell ^{-1}}]f\Vert _0&\lesssim \ell \Vert f\Vert _0\Vert \nabla g\Vert _0 \\ \Vert [g, P_{\leqslant \ell ^{-1}}]f\Vert _N&\lesssim _N \ell ^{1-N} \Vert f\Vert _0\Vert g\Vert _{\max \{1,N\}}. \end{aligned}\end{aligned}$$

(A.3)

In particular, for any smooth function $v, F\in C^\infty (\mathbb {T}^3)$ and for each $N\geqslant 0$, we have

$$\begin{aligned}&\Vert [v\cdot \nabla , {P}_{\leqslant \ell ^{-1}}]F\Vert _0=\Vert [v\cdot \nabla , {P}_{> \ell ^{-1}}] F\Vert _{0} \lesssim \ell \Vert \nabla F\Vert _0\Vert \nabla v\Vert _0 \end{aligned}$$

(A.4)

$$\begin{aligned}&\Vert [v \cdot \nabla , {P}_{\leqslant \ell ^{-1}}]F\Vert _N =\Vert [v\cdot \nabla , {P}_{> \ell ^{-1}}] F\Vert _{N}\lesssim _N \ell ^{1-N}\Vert \nabla F\Vert _0\Vert v\Vert _{\max \{1,N\}}. \end{aligned}$$

(A.5)

Remark A.5

When v has the frequency localized to $\ell ^{-1}$, using the Bernstein’s inequality, (A.5) can be improved to $\Vert [v\cdot \nabla , {P}_{\leqslant \ell ^{-1}}]F\Vert _N\lesssim _N \ell ^{1-N}\Vert \nabla v\Vert _0\Vert \nabla F\Vert _0$.

Proof

We first write

Then, (A.3) is obtained as $\Vert f_\ell g - (fg)_\ell \Vert _0 \leqslant \ell \Vert f\Vert _0\Vert \nabla g\Vert _0$ and

for some constants $c_{N_1,N}$. Since we have

$$\begin{aligned}{}[v\cdot \nabla , {P}_{> \ell ^{-1}}] F (x)&= v\cdot \nabla ({P}_{> \ell ^{-1}}F -F)+(v\cdot \nabla F) - {P}_{> \ell ^{-1}}(v\cdot \nabla F) \\&= -v \cdot \nabla {P}_{\leqslant \ell ^{-1}} F + {P}_{\leqslant \ell ^{-1}} (v \cdot \nabla F) = - [v\cdot \nabla , {P}_{\leqslant \ell ^{-1}}] F, \end{aligned}$$

we apply (A.3) to $g=v_i$ and $f=\partial _i F$, then (A.4) and (A.5) follow. $\square$

Lemma A.6

For a fixed $\overline{N}\in {\mathbb {N}}$, if v and g satisfy

$$\begin{aligned} \Vert v\Vert _N \lesssim _{{\overline{N}}} \ell ^{-N}v_F, \quad \Vert g\Vert _N \lesssim _{{\overline{N}}} g_F \end{aligned}$$

for all integer $N\in [1, \overline{N}]$ and for some positive constants $v_F$ and $g_F$, then we have

$$\begin{aligned} \Vert [v \cdot \nabla , P_{\leqslant \ell ^{-1}}](fg)-([v \cdot \nabla , P_{\leqslant \ell ^{-1}}]f) g\Vert _{N} \lesssim \ell ^{1-N}\Vert \nabla f\Vert _0 v_F g_F +\ell ^{-N}\Vert f\Vert _0v_Fg_F, \end{aligned}$$

(A.6)

for any integer $N\in [0,{\overline{N}}]$.

Proof

We first write

Then, (A.6) follows from

$\square$

Lemma A.7

For vector-valued functions H and v in $C^\infty ([0,T]\times \mathbb {T}^3)$, it holds that

$$\begin{aligned} \Vert [P_{\lesssim \ell ^{-1}}v \cdot \nabla , {\mathcal {R}}]P_{\gtrsim \lambda _{q+1}} H\Vert _{N-1} \lesssim \sum _{N_1+N_2=N-1} \ell \Vert \nabla v\Vert _{N_1} \Vert H\Vert _{N_2} \end{aligned}$$

for $N=1,2$, where ${\mathcal {R}}= \Delta ^{-1}\nabla$.

Proof

For convenience, we write $v_\ell := P_{\lesssim \ell ^{-1}}v$ and $H_j = P_{2^j}H$ for for $2^j \gtrsim \lambda _{q+1}$. We first use the Fourier expansion and the Taylor’s theorem to get

$$\begin{aligned}&{-}[v_{\ell } \cdot \nabla , {\mathcal {R}}]H_j \nonumber \\&\quad = \sum _{k, \eta \in \mathbb {Z}^3} ({\mathscr {F}}[{\mathcal {R}}](k)-{\mathscr {F}}[{\mathcal {R}}](\eta )) i\eta \cdot {\mathscr {F}}[ v_\ell ](k-\eta ) {\mathscr {F}}[H_j] (\eta ) e^{ik\cdot x}\nonumber \\&\quad = \sum _{k, \eta \in \mathbb {Z}^3} \sum _{l=1}^{l_0} \frac{1}{l!}[(k-\eta )\cdot \nabla ]^l {\mathscr {F}}[{\mathcal {R}}](\eta ) i\eta \cdot {\mathscr {F}}[v_\ell ](k-\eta ) {\mathscr {F}}[H_j] (\eta ) e^{ik\cdot x}\nonumber \\&\begin{aligned}&\qquad + \frac{1}{l_0!} \sum _{k, \eta \in \mathbb {Z}^3} \int _0^1 [(k-\eta )\cdot \nabla ]^{l_0+1} {\mathscr {F}}[{\mathcal {R}}](\eta + \sigma (k-\eta ))(1-\sigma )^{l_0} d\sigma i\eta \\&\qquad \cdot {\mathscr {F}}[ v_\ell ](k-\eta ) {\mathscr {F}}[H_j] (\eta ) e^{ik\cdot x}, \end{aligned} \end{aligned}$$

(A.7)

where $l_0>2$, independent of q, is chosen to satisfy $(\ell ^{-1}\lambda _{q+1}^{-1})^{l_0} \lambda _{q+1}^3\lesssim 1$. The first term can be written as $\sum _{l=1}^{l_0}\frac{1}{l!} \nabla ^l v_\ell : \nabla {\mathcal {R}}_l H_j$ where the operator ${\mathcal {R}}_l$ has a Fourier multiplier defined by ${\mathscr {F}}[{\mathcal {R}}_l g](\eta ) = { (-i)^l}\nabla _\eta ^l {\mathscr {F}}[{\mathcal {R}}](\eta ) {\mathscr {F}}[g](\eta )$.

Using this decomposition, we now estimate

$$\begin{aligned}&\left\| \sum _{l=1}^{l_0}\frac{1}{l!} \nabla ^l v_\ell : \nabla {\mathcal {R}}_l P_{\gtrsim \lambda _{q+1}} H\right\| _{N-1} \\&\quad \lesssim \sum _{N_1+N_2=N-1} \sum _{l=1}^{l_0}\frac{1}{l!} \Vert \nabla ^l v_\ell \Vert _{N_1}\Vert \nabla {\mathcal {R}}_l P_{\gtrsim \lambda _{q+1}} H\Vert _{N_2}\\&\quad \lesssim \sum _{N_1+N_2=N-1} \sum _{l=1}^{l_0}\frac{\ell ^{1-l}}{l!} \Vert \nabla v \Vert _{N_1} \sum _{2^j\gtrsim \lambda _{q+1}}\Vert K_{l,j}\Vert _{L^1} \Vert H\Vert _{N_2}\\&\quad \lesssim \sum _{N_1+N_2=N-1} \ell \Vert \nabla v \Vert _{N_1} \Vert H\Vert _{N_2}, \end{aligned}$$

where $K_{l,j}$ is the kernel of the operator $\nabla {\mathcal {R}}_l P_{2^j}$ and the last estimate follows from

$$\begin{aligned} K_{l,j} = 2^{j(-l+3)}K_{l,0}(2^j \cdot ), \quad \sum _{2^j\gtrsim \lambda _{q+1}}\Vert K_{l,j}\Vert _{L^1(\mathbb {R}^3)} \lesssim \lambda _{q+1}^{-l} \Vert K_{l,0}\Vert _{L^1(\mathbb {R}^3)}. \end{aligned}$$

The remaining term can be estimated as follows: since $|k-\eta |\leqslant \frac{1}{2} |\eta |$ and hence $|k|\lesssim |\eta |$, we get

$$\begin{aligned} \left\| \nabla ^{N-1} \sum _{2^j\gtrsim \lambda _{q+1}} (A.7)\right\| _0&\lesssim {\sum _{2^j\gtrsim \lambda _{q+1}}}\sum _{k, \eta \in \mathbb {Z}^3} |k|^{N-1} \frac{|k-\eta |^{l_0+1}}{|\eta |^{l_0+1}} |{\mathscr {F}}[v_\ell ](k-\eta )| |{\mathscr {F}}[H_j] (\eta )|\\&\lesssim \sum _{2^j\gtrsim \lambda _{q+1}}\sum _{k, \eta \in \mathbb {Z}^3} \frac{|k-\eta |^{l_0}}{|\eta |^{l_0+1}} |{\mathscr {F}}[\nabla v_\ell ](k-\eta )| |\eta |^{N-1}|{\mathscr {F}}[H_j] (\eta )|\\&\lesssim \sum _{2^j\gtrsim \lambda _{q+1}} \sum _{|k|\lesssim 2^j} (\ell ^{-1}2^{-j})^{l_0} 2^{-j} \Vert \nabla v\Vert _{L^2(\mathbb {T}^3)}\Vert {\nabla ^{N-1}}H_j\Vert _{L^2(\mathbb {T}^3)}\\&\lesssim \sum _{2^j\gtrsim \lambda _{q+1}} (\ell ^{-1}2^{-j})^{l_0} 2^{2j} \Vert \nabla v\Vert _{L^2(\mathbb {T}^3)}\Vert \nabla ^{N-1}H_j\Vert _{L^2(\mathbb {T}^3)}\\&\lesssim (\ell ^{-1}\lambda _{q+1}^{-1})^{l_0} \lambda _{q+1}^2 \Vert \nabla v\Vert _0 \Vert H\Vert _{N-1} \lesssim \lambda _{q+1}^{-1}\Vert \nabla v\Vert _0 \Vert H\Vert _{N-1}. \end{aligned}$$

$\square$