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

Šárka Nečasová (1) , Xavier Blanc (2, 3) , Raphaël Danchin (4) , Bernard Ducomet (4)

1 IM / CAS - Institute of Mathematics of the Czech Academy of Science
2 UPD7 - Université Paris Diderot - Paris 7
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Mots clés: compressible Euler system; Helmholtz; Poisson; global solution; decay AMS subject classification: 35Q30; 76N10

Domaines: Equations aux dérivées partielles [math.AP]

Šárka Nečasová, Xavier Blanc, Raphaël Danchin, Bernard Ducomet. The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings. Journal of Hyperbolic Differential Equations, 2021, 18 (1), pp.169-193. ⟨10.1142/S0219891621500041⟩. ⟨hal-02160016⟩