On an Euler–Schrödinger system appearing in laser-plasma interaction

We consider the Cauchy problem for the barotropic Euler system coupled to a vector Schrödinger equation in the whole space. Assuming that the initial density and vector potential are 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 bounded away from zero, we prove the existence of a global-in-time unique solution with (fractional) Sobolev regularity. Moreover, we obtain some algebraic time decay estimates of the solution. Our work extends the papers by D. Serre and M. Grassin [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C.R. Acad. Sci. Paris, Série I 325 (1997) 721–726; 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] and previous works by B. Ducomet and co-authors [The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings, J. Hyper. Differ. Equ. 18(1) (2021) 169–193; On the the global existence for the compressible Euler-Poisson system, and the instability of static solutions, J. Evol. Equ. 21(3) (2021) 3035–3054] dedicated to the compressible Euler–Poisson system.