Global existence of smooth solutions for the magnetic Schrödinger equation arising from hot plasma (Q738726)

The paper deals with the following system \[ iE_t+\Delta E+|E|^2E+iE\times B=0, \] \[ \Delta B-i\beta \nabla\times(\nabla\times(E\times{\bar E}))-\gamma\partial_t\int_{\mathbb{R}^3}\frac{B(t,y)}{|x-y|^2}dy=0, \] which arises in the infinite ion acoustic speed limit of the magnetic Zakharov system in a hot plasma. It is considered the initial value problem with initial conditions \(E(0,x)=E_0(x)\) and \(B(0,x)=B_0(x)\). The main result is the following. Let \(N\geq 100\) and \(0<\delta\leq 10^{-3}\). There exists \(\epsilon_0\in (0, 1)\) such that if \[ \|E_0\|_{H^N}+\|xE_0\|_{L^2}+\||x|^2E_0\|\leq\epsilon_0, \] \[ \|B_0\|_{H^{N-1}}+\|B_0\|_{L^1}\leq\epsilon_0, \] then the initial value problem has a unique global solution \((E,B)\in C(\mathbb{R}_+; H^N\times H^{N-1})\) satisfying \[ \|E(t)\|_{H^N}\lesssim \epsilon (1+t)^\delta,\quad \|B(t)\|_{H^{N-1}}\lesssim\epsilon_0, \] \[ \|E(t)\|_{L^\infty}\lesssim\frac{\epsilon_0}{(1+t)^{7/6-\delta}},\quad \|B(t)\|_{L^\infty}\lesssim\frac{\epsilon_0}{1+t}. \] The \(L^\infty\) dacay rate above is optimal.