  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.
