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

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 [Formula: see text] such that the spectrum of [Formula: see text] is positive and 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 Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

Low Mach and thin domain limit for the compressible Euler system

“…We would like to stress that, under certain hypotheses on initial data, Serre and Grassin [27,28] proved global smooth solutions to the compressible Euler in R d (d ≥ 1). Recently, these results have been extended by Blanc et al [4] for the Euler system coupled to the Helmholtz or Poisson equations.…”

The pressureless damped Euler–Riesz equations

“…As for the compressible Euler equations with Riesz interactions, Choi [14] proved the local well-posedness of classical solutions and exhibited sufficient conditions of finite time blow-up phenomena. We also mention that Danchin and Ducomet [23,24,25] dealt with the well-posedness issue in the case that the density contains vacuum and that the initial velocity allows some reference vector field. For the one-dimensional pressureless Euler flows with non-local forces, there exist critical thresholds between the subcritical region with global regularity and the supercritical region with finite-time blow-up of classical solutions, cf.…”