The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

2019/06/19 by Šárka Nečasová, Nečasová, Šárka, Xavier Blanc +6 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1906.08075

openalex publication_date 2019/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u0 such that the spectrum of Du0 is bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends the papers by D. Serre and M. Grassin [20, 21, 43] dedicated to the compressible Euler system without coupling and integer regularity exponents.

Citations

• The Cauchy problem for quasi-linear symmetric hyperbolic systems
• Nonlinear scalar field equations, II existence of infinitely many solutions
• Nonlinear scalar field equations, I existence of a ground state

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