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.

我们在整个空间中考虑与Poisson方程耦合的正压Euler系统的Cauchy问题。我们的主要目的是展示一个简单的功能框架，该框架可以处理无穷大密度为零的解决方案，但无需紧凑地支持。当多方索引\（\ gamma \）时，我们尤其要牢记3D静态解的气体等于6/5。我们的第一个结果是在简单的功能框架中经典解的局部存在，不需要速度在无穷大时趋于0，并且不需要紧凑地支持密度。接下来，紧随Grassin和Serre致力于可压缩的Euler系统Grassin和Serre的工作（CR Acad Sci ParisSérI 325：721–726，1997，Grassin（印第安纳大学数学学报，47：1397–1432，1998）），我们表明如果初始密度足够小，并且初始速度接近某个参考矢量场\（u_0 \）从而\（Du_0 \）的光谱是正数且远离零有界，则相应的经典解是全局的，并且满足代数时间衰减估计。与我们最近的论文（Blanc等人在“具有Poisson或Helmholtz耦合的可压缩Euler系统的全球存在性问题”中）相比，我们能够处理上述3D静态解，并在功能范围内证明其不稳定性。框架。