On an Euler-Schrödinger system appearing in laser-plasma interaction
Kuntal Bhandari, Bernard Ducomet, Šarka Nečasová, John Sebastian H. Simon
TL;DR
The paper studies a global well-posedness problem for the compressible Euler system coupled to a vector Schrödinger equation in ${\mathbb R}^d$, under small initial density and a spectral gap condition on the initial velocity. Using Makino's symmetrization, an auxiliary Burgers flow, and Sobolev-energy methods, the authors prove global existence and uniqueness of solutions near the trivial state, with the solution components $(\varrho^{(\gamma-1)/2},\,u-v,\,A)$ remaining in $C(\mathbb{R}_+; H^s)$ for suitable $s$. They also establish algebraic decay in Sobolev norms: for $0\le\sigma\le s$, $\|\varrho^{(\gamma-1)/2},\,u-v,\,A\|_{\dot{H}^\sigma} \lesssim (1+t)^{d/2-\sigma-\min(1, d(\gamma-1)/2)}$. The analysis blends dispersion from the Burgers flow with the Schrödinger coupling and develops decay estimates that underpin global stability of the trivial state.
Abstract
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 $u_0$ such that the spectrum of $Du_0$ 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 [11, 13, 19] and previous works by B. Ducomet and co-authors [4, 8] dedicated to the compressible Euler-Poisson system.