On the global existence for the compressible Euler–Poisson system, and the instability of static solutions

Abstract

We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index $\gamma$ of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Sér I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field $u_0$ such that the spectrum of $Du_0$ is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

Appendix

For the reader’s convenience, we here recall some technical results that have been used in the paper. More details may be found in the appendix of [4].

The first result is a differential inequality that leads to a control of the solution for all positive times.

Lemma 3.1

Let $Y:\mathbb {R}_+\rightarrow \mathbb {R}_+$ satisfy the differential inequality

$$\begin{aligned} \frac{\text {d}}{{\text {d}}t}Y+\frac{a}{1+t}Y\le C\biggl (\frac{Y}{(1+t)^2}+Y^2+(1+t)^{m'-1}Y^{m+1}\biggr )\quad \hbox {on }\ \mathbb {R}_+ \end{aligned}$$

for some $C>0,$ $a>1,$ $m>0$ and $m'<ma.$ Then, there exists $c=c(a,m,m',C)$ such that if $Y(0)\le c,$ then we have

$$\begin{aligned} Y(t)\le 2e^{\frac{Ct}{1+t}}\frac{Y_0}{(1+t)^a}\quad \hbox {for all }\ t\ge 0. \end{aligned}$$

The following result (see [4, 17]) has been used to bound the potential term.

Lemma 3.2

Let $\alpha \ge 1$ and $0\le \sigma <\alpha +\frac{1}{2}$. Then, we have the inequality

$$\begin{aligned} \Vert |z|^\alpha \Vert _{\dot{H}^\sigma }\lesssim \Vert z\Vert _{L^\infty }^{\alpha -1}\Vert z\Vert _{\dot{H}^\sigma }. \end{aligned}$$

(3.20)

We also used the following first-order commutator estimate that corresponds to the end of [20, Rem. 1.5], or may be seen as a straightforward adaptation to the homogeneous framework of the second inequality of [2, Lemma A.2]:

Lemma 3.3

If $s>0,$ then we have:

$$\begin{aligned} \Vert [v,\dot{\Lambda }^s]u\Vert _{L^2}\lesssim \Vert v\Vert _{\dot{H}^s}\Vert u\Vert _{L^\infty }+\Vert \nabla v\Vert _{L^\infty }\Vert u\Vert _{\dot{H}^{s-1}}. \end{aligned}$$

The following second-order commutator inequality that has been established in [4] (after a prior result in [2]) enabled us to prove the $\dot{H}^s$ estimate for the solutions to (3.2).

Lemma 3.4

If $s>1$ then, we have:

$$\begin{aligned} \Vert [v,\dot{\Lambda }^s]u-s\nabla v\cdot \dot{\Lambda }^{s-2}\nabla u\Vert _{L^2}\lesssim \Vert v\Vert _{\dot{H}^s}\Vert u\Vert _{L^\infty }+\Vert \nabla ^2v\Vert _{L^\infty }\Vert u\Vert _{\dot{H}^{s-2}}. \end{aligned}$$