Expanding large global solutions of the equations of compressible fluid mechanics

Abstract

Without any symmetry assumptions on the initial data we construct global-in-time unique solutions to the vacuum free boundary three-dimensional isentropic compressible Euler equations when the adiabatic exponent $\gamma$ lies in the interval $(1,\frac{5}{3}]$. Our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by Sideris and they satisfy the physical vacuum boundary condition.

Appendices

A Asymptotic-in-$\tau$ behavior of affine solutions

In this section we collect some of the basic properties of affine motions that are used at many places in our estimates. Their proofs are rather straightforward and follow directly from the description of the asymptotic behavior of the solutions of (2.18) from [44].

We remind the reader that for any $M\in {\mathbb {M}}^{3\times 3}$ we denote by $\Vert M\Vert$ the Hilbert–Schmidt norm of the matrix M.

Lemma A.1

(Asymptotic behavior of A, $\Gamma ^*= O^{-1} O_\tau$, and $\Lambda = O^{-1} O^{-\top }$) For any $\gamma \in (1,\frac{5}{3}]$ and any pair of initial conditions $(A(0),\dot{A}(0))\in {\mathrm{GL}}^+(3)\times {\mathbb {M}}^{3\times 3}$ there exist matrices $A_0,A_1,M(t)$ such that the unique solution A(t) to the Cauchy problem

$$\begin{aligned} A_{tt}&= \delta \det {A}^{1-\gamma } A^{-\top } \end{aligned}$$

(A.1)

$$\begin{aligned} A(0)&= A(0), \quad A_t(0) = \dot{A}(0) \end{aligned}$$

(A.2)

can be written in the form

$$\begin{aligned} A(t) = A_0 + t A_1 + M(t), \quad t\ge 0, \end{aligned}$$

(A.3)

where $A_0,A_1$ are both time-independent and M(t) satisfies the bounds

$$\begin{aligned} \Vert M(t)\Vert = o_{t\rightarrow \infty }(1+t), \quad \Vert \partial _t M(t)\Vert \lesssim (1+t)^{3-3\gamma }. \end{aligned}$$

(A.4)

Moreover

$$\begin{aligned} e^{\mu _1\tau } \lesssim \mu (\tau ) \lesssim e^{\mu _1\tau }, \quad \tau \ge 0, \end{aligned}$$

(A.5)

$$\begin{aligned} \lim _{\tau \rightarrow \infty } \Vert \Gamma ^* - e^{-\mu _1\tau } \mu _1 A_0 A_1^{-1} \Vert = 0, \end{aligned}$$

(A.6)

where

$$\begin{aligned} \Gamma ^* = O^{-1} O_\tau , \quad \mu _1 = (\det A_1)^{\frac{1}{3}}>0. \end{aligned}$$

Furthermore there exists a constant $C>0$ such that

$$\begin{aligned}&\Vert \Lambda _\tau \Vert \le C e^{-\mu _1\tau }, \quad \Vert \Lambda _{\tau \tau }\Vert \le C e^{-2\mu _0\tau } , \quad \Vert \Lambda \Vert + \Vert \Lambda ^{-1}\Vert \le C, \end{aligned}$$

(A.7)

$$\begin{aligned}&\sum _{i=1}^3\left( d_i+\frac{1}{d_i}\right) \le C \end{aligned}$$

(A.8)

$$\begin{aligned}&\sum _{i=1}^3|\partial _\tau d_i| + \Vert \partial _\tau P\Vert \le C e^{-\mu _1\tau } \end{aligned}$$

(A.9)

$$\begin{aligned}&\frac{1}{C}|\mathbf{w}|^2 \le \langle \Lambda ^{-1}{} \mathbf{w}, \mathbf{w}\rangle \le C |\mathbf{w}|^2 , \ \mathbf{w}\in {\mathbb {R}}^3, \end{aligned}$$

(A.10)

where $d_i$, $i=1,2,3$, are the eigenvalues of the matrix $\Lambda$ and $P\in \text {SO}(3)$ satisfies $\Lambda = P^\top Q P,$ $Q = {\mathrm{diag}}(d_i)$.

Proof

Asymptotic behavior (A.3)–(A.4) and bound (A.5) are a consequence of Theorem 3 and Lemma 6 from [44].

Proof of (A.6) Since $O = \frac{A}{\mu }$ we have

$$\begin{aligned} O_t = \frac{A_t}{\mu } - \frac{A\mu _t}{\mu ^2} = \frac{A_t}{\mu } - \frac{A\text {Tr}(A^{-1}A_t)}{3\mu }, \end{aligned}$$

(A.11)

where we used the formula

$$\begin{aligned} \mu _t {=}\partial _t\left( (\det A)^{\frac{1}{3}}\right) {=} \frac{1}{3} \mu ^{-2} \partial _t\det A = \frac{1}{3} \mu ^{-2} \mu ^3 \text {Tr}(A^{-1}A_t) = \frac{1}{3} \mu \text {Tr}(A^{-1}A_t). \end{aligned}$$

Therefore, from (A.3) and (A.4) it is easy to obtain the following asymptotics:

$$\begin{aligned} \text {Tr}(A^{-1}A_t)&\sim \text {Tr}((A_0+tA_1 + M(t))^{-1}(A_1+\partial _t M(t))) \nonumber \\&=t^{-1}\text {Tr}\left( \left( \frac{A_0}{t}+A_1\right) ^{-1}\left( A_1+O\left( t^{3-3\gamma }\right) \right) \right. \nonumber \\&\sim t^{-1}\text {Tr}\left( A_1^{-1}\left( A_1+O\left( t^{3-3\gamma }\right) \right) = \frac{3}{t} + O(t^{2-3\gamma }). \right. \end{aligned}$$

(A.12)

This implies that

$$\begin{aligned} O_t\sim \frac{A_1}{\mu } - \frac{(A_0+tA_1)\frac{3}{t}}{3\mu } = \frac{A_0}{t\mu }, \end{aligned}$$

where we made use of (A.4) again. Recalling that $\frac{d\tau }{dt} = \frac{1}{\mu }$ we obtain the following asymptotic behavior $O_\tau \sim _{\tau \rightarrow \infty } \frac{A_0}{t(\tau )}.$ Using (A.3) again it follows that

$$\begin{aligned} \det A\sim \det (A_0+t A_1) = t^3 \det \left( \frac{A_0}{t} + A_1\right) \sim t^3\det A_1, \quad \mu \sim t (\det A_1)^{\frac{1}{3}}. \end{aligned}$$

(A.13)

Since

$$\begin{aligned} t = t(\tau ) \sim _{\tau \rightarrow \infty } e^{(\det A_1)^{\frac{1}{3}}\tau }, \end{aligned}$$

and $O \sim \frac{A_0+tA_1}{\mu }$ we conclude that

$$\begin{aligned} O^{-1}\sim \frac{\mu }{t} \left( \frac{A_0}{t}+A_1\right) ^{-1}\sim \frac{\mu }{t} A_1^{-1} \sim (\det A_1)^{\frac{1}{3}}A_1^{-1}, \end{aligned}$$

where we used (A.13). Therefore

$$\begin{aligned} \Gamma ^* = O^{-1} O_\tau \sim (\det A_1)^{\frac{1}{3}}A_1^{-1}\frac{A_0}{t(\tau )} = e^{-\mu _1\tau } \mu _1 A_0 A_1^{-1}, \end{aligned}$$

where $\mu _1 = (\det A_1)^{\frac{1}{3}},$ and this completes the proof of (A.6). Proof of (A.7) is similar.

Proof of (A.7)–(A.9) From the definition of $\Lambda$ we have

$$\begin{aligned} \Lambda _\tau&= - O^{-1} O_\tau O^{-1} O^{-\top } - O^{-1} O^{-\top } O^\top _\tau O^{-\top } \nonumber \\&= -\Gamma ^*\Lambda - \Lambda (\Gamma ^*)^\top = -2 \Lambda (\Gamma ^*)^\top , \end{aligned}$$

(A.14)

where we used the symmetry of $\Lambda$ in the last equality. Since $\Vert \Lambda \Vert \lesssim 1$ it follows by part (i) that $\Vert \Lambda _\tau \Vert \lesssim e^{-\mu _1\tau }$. To bound $\Lambda _{\tau \tau }$ we note that by (A.14) $\Lambda _{\tau \tau } = - 2 \Lambda _\tau (\Gamma ^*)^\top - 2\Lambda (\Gamma _\tau ^*)^\top .$ Since both $\Vert \Lambda _\tau \Vert$ and $\Vert \Gamma ^*\Vert$ decay exponentially, it remains to prove the decay of $\Vert \Gamma _\tau ^*\Vert$. From $\Gamma ^* = O^{-1}O_\tau$ it follows that $\Gamma _\tau ^*=-(\Gamma ^*)^2 + O^{-1}O_{\tau \tau }$, and therefore it remains to prove the decay of $\Vert O_{\tau \tau }\Vert$. A simple calculation shows that

$$\begin{aligned} O_{\tau \tau } = \mu \left( A_{tt} - \frac{A_t \text {Tr}(A^{-1}A_t)+A \partial _t\text {Tr}(A^{-1}A_t)}{3} \right) . \end{aligned}$$

Using the asymptotic behavior (A.12), (2.18), we can refine the asymptotics (A.12) to show that $A_t \text {Tr}(A^{-1}A_t)+A \partial _t\text {Tr}(A^{-1}A_t) = O(t^{2-3\gamma })$ and therefore from the above equation it follows that

$$\begin{aligned} \Vert O_{\tau \tau }\Vert \lesssim (1+t)^{3-3\gamma } \lesssim e^{-2\mu _0\tau }. \end{aligned}$$

This yields the second bound in (A.7). Bounds (A.8) (A.9) follow by similar arguments using (A.3), while (A.10) is a direct consequence of (A.8). $\square$

B Commutators

In order to evaluate various commutator terms that arise from commuting differential operators with the usual Cartesian derivatives or apply the Leibniz rule we shall rely on the fact that the high-order Sobolev norms expressed in polar coordinates are equivalent to the usual high-order Sobolev norms on the support of function $\psi$.

Lemma B.1

Let be a collection of the standard Cartesian, normal, and tangential vector-fields. For any two vector fields $X_k,X_\ell \in {\mathcal {X}}$, $k,\ell = 1,\ldots ,7$ there commutator satisfies the following relationship

where the functions $c^m_{k\ell }$, $c_{k\ell }$, are $C^\infty$ on the exterior of any ball around the origin $r=0$.

Proof

The proof is a simple consequence of the following direct calculations. For any $i,j\in \{1,2,3\}$ we have

(B.1)

$\square$

Lemma B.1 is a technical tool allowing us to bound the lower order commutators in our energy estimates.

Using the product rule and the relationship $\mathscr {A}=[D\eta ]^{-1}$ it is easy to see that the following formulas hold

$$\begin{aligned} \partial _r\mathscr {A}^k_i&= -\mathscr {A}^k_s\partial _r\partial _m\eta ^s\mathscr {A}^m_i = -\mathscr {A}^k_s\partial _r\left( \partial _m\uptheta ^s+\delta ^s_m\right) \mathscr {A}^m_i \nonumber \\&= -\mathscr {A}^k_s(\partial _r\uptheta ^s),_m\mathscr {A}^m_i + \mathscr {A}^k_s [\partial _m,\partial _r]\uptheta ^s\mathscr {A}^m_i. \end{aligned}$$

(B.2)

Similarly, for any $j=1,2,3$ we have

(B.3)

Therefore, if $\beta =(0,0,0)$ we have the formula

(B.4)

If $|\beta | > 0$, then with $e_1=(1,0,0)$, $e_2 = (0,1,0)$, and $e_3=(0,0,1)$ we obtain

(B.5)

where $c_{a'\beta '},c_{\beta '}$ are positive universal constants and the Einstein summation convention does not apply to the index j. Finally, the high-order commutators appearing on the right-hand side in the identities (B.4)–(B.5) can be expressed as a linear combination of the elements of ${\mathcal {X}}$ with smooth coefficients away from zero. In other words, for any $j=1,2,3$ the following commutator identity holds:

(B.6)

for some universal coefficients $C^j_{a',\beta ',\ell }$ which are smooth on the exterior of any ball around the origin $r=0$. Formula (B.6) is a direct consequence of Lemma B.1.

C Weighted spaces, Hardy inequalities, and Sobolev embeddings

In this section, we recall Hardy inequalities and embedding results of weighted function spaces. First of all, we state the Hardy inequality near $r=1$.

Lemma C.1

(Hardy inequality [20]) Let k be a real number and g a function satisfying $\int _0^1 (1-r)^{k +2}(g^2 + g'^2) dr < \infty$.

If $k > -1$, then we have $\int _0^1 (1-r)^{k} g^2 dr \le C \int _0^1 (1-r)^{k+2} (g^2 + |g'|^2) dr$.

If $k < -1$, then g has a trace at $r=1$ and $\int _0^1 (1-r)^{k} (g - g(1))^2 dr \le C \int _0^1 (1-r)^{k+2} |g'|^2 dr$.

Since w depends only on r and w behaves like a distance function $1-r$, using Lemma C.1, we in particular get

$$\begin{aligned} \int _{B_1(\mathbf{0}){\setminus } B_{\frac{1}{4}}(\mathbf{0})} \psi w^{k} | u |^2 dy \, \lesssim \int _{B_1(\mathbf{0}){\setminus } B_{\frac{1}{4}}(\mathbf{0})} \psi w^{k+2} \left( |\partial _r u |^2 + |u|^2 \right) dy \end{aligned}$$

(C.1)

for any nonnegative real number $k> -1$ and for any $u\in C^\infty (B_1(\mathbf{0}){\setminus } B_{\frac{1}{4}}(\mathbf{0}))$. We can apply (C.1) to and thereafter apply (C.1) repeatedly to the right-hand side. As a consequence for any $-1< k < \alpha +a$,

(C.2)

Upon choosing $m = \lceil a+\alpha -k \rceil$ where $\lceil \ \rceil$ is the ceiling function, we obtain

(C.3)

for any $u\in C^\infty (B_1(\mathbf{0}){\setminus } B_{\frac{1}{4}}(\mathbf{0}))$.

As a consequence of (C.3), we obtain the weighted Sobolev–Hardy inequality:

Proposition C.2

For any $u\in C^\infty (B_1(\mathbf{0}))$, we have

(C.4)

(C.5)

We omit the technical details of the proof, as it follows from standard estimates relying on the $H^2(\Omega )\hookrightarrow L^\infty (\Omega )$ continuous embedding and the Hardy inequality (C.3).

D Starting from Lagrangian formulation

Consider the Lagrangian formulation of the E$_\gamma$-system [6, 19]:

$$\begin{aligned} \zeta _{tt} + \tfrac{\gamma }{\gamma -1}\mathscr {A}_\zeta ^T \nabla (w {\mathscr {J}}_\zeta ^{1-\gamma })=0 \end{aligned}$$

(D.1)

where $\mathscr {A}_\zeta$ and ${\mathscr {J}}_\zeta$ are induced by the flow map $\partial _t\zeta (t,y)=\mathbf{{u}}(t,\zeta (t,y))$ ($\mathbf{u}$ is the original fluid velocity appearing in (1.1)) and $w=(\rho _0 {\mathscr {J}}_\zeta (0))^{\gamma -1}\ge 0$ is an enthalpy function that depends only on the initial data. To find affine motions (with center of mass at 0) one makes the ansatz $\zeta (t,y)= A(t) y$ [44]. With $\mathscr {A}_{\zeta }^\top =[D\zeta ]^{-\top }= A(t)^{-\top }$ and ${\mathscr {J}}_{\zeta }=\det A$, by plugging this ansatz into (D.1), we obtain

$$\begin{aligned} A_{tt} y + \tfrac{\gamma }{\gamma -1} (\det A)^{1-\gamma } A^{-\top } \nabla w=0 \end{aligned}$$

(D.2)

Since w is independent of t, (D.2) will be satisfied if we demand

$$\begin{aligned} A_{tt}= \delta (\det A)^{1-\gamma } A^{-\top } , \quad \delta y = - \tfrac{\gamma }{\gamma -1} \nabla w, \quad \delta >0, \end{aligned}$$

(D.3)

which precisely yields the affine motions (1.5)–(1.6) (discovered by Sideris when $\mathbf{a}\equiv 0$) which form the set ${\mathscr {S}}$ with the center of mass fixed at the origin.

In order to study the stability of elements of ${\mathscr {S}}$ we want to realize them as time-independent background solutions. Given such an affine motion A we modify the flow map $\zeta$ and define $\eta := A^{-1} \zeta$. Since $\mathscr {A}_\zeta ^\top = A^{-\top } \mathscr {A}_\eta ^\top$ and ${\mathscr {J}}_\zeta = (\det A) {\mathscr {J}}_\eta$, we obtain

$$\begin{aligned} \eta _{tt} + 2 A^{-1} A_t \eta _t + A^{-1} A_{tt}\eta + \tfrac{\gamma }{\gamma -1} (\det A)^{1-\gamma } A^{-1} A^{-\top } \mathscr {A}_\eta ^\top \nabla (w {\mathscr {J}}_\eta ^{1-\gamma }) =0 \end{aligned}$$

By using (D.3) we can rewrite the previous equation in the form

$$\begin{aligned} (\det A)^{\gamma -\frac{1}{3}} \eta _{tt} + 2 (\det A)^{\gamma -\frac{1}{3}} A^{-1} A_t \eta _t +\delta \Lambda \eta + \tfrac{\gamma }{\gamma -1} \Lambda \mathscr {A}_\eta ^\top \nabla (w {\mathscr {J}}_\eta ^{1-\gamma }) =0 \end{aligned}$$

(D.4)

where $\Lambda =\det A^{\frac{2}{3}} A^{-1}A^{-T}$.

We rescale the time variable t so that $1+t\sim e^{\mu _1\tau }$ by setting $\frac{d\tau }{d t} = (\det A)^{-\frac{1}{3}}$. Then (D.4) can be written as

$$\begin{aligned}&(\det A)^{\gamma -1} \eta _{\tau \tau } -\tfrac{1}{3} (\det A)^{\gamma -2} (\det A)_\tau \eta _\tau + 2 (\det A)^{\gamma -1} A^{-1} A_\tau \eta _\tau \nonumber \\&\quad +\delta \Lambda \eta + \tfrac{\gamma }{\gamma -1} \Lambda \mathscr {A}_\eta ^\top \nabla (w {\mathscr {J}}_\eta ^{1-\gamma }) =0 \end{aligned}$$

(D.5)

We now recall $A^{-1} A_\tau = \mu ^{-1} \mu _\tau I + O^{-1}O_\tau$ where $A= \mu O$ and $\mu = (\det A)^{\frac{1}{3}}$ (see Sect. 2.1). The equation for $\eta$ reads

$$\begin{aligned} \mu ^{3\gamma -3}\eta _{\tau \tau } + \mu ^{3\gamma -4}\mu _\tau \eta _\tau + 2 \mu ^{3\gamma -3}\Gamma ^*\eta _\tau +\delta \Lambda \eta + \tfrac{\gamma }{\gamma -1} \Lambda \mathscr {A}_\eta ^\top \nabla (w {\mathscr {J}}_\eta ^{1-\gamma }) =0 \end{aligned}$$

(D.6)

where $\Gamma ^*=O^{-1}O_\tau$. It is clear that $\eta (y)\equiv y$ corresponds to Sideris’ affine motions, and equation (D.6) is nothing but (2.35). By considering $\uptheta =\eta -y$, we obtain the $\uptheta$-Eq. (2.38).