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

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