Global well-posedness below energy space for the \(1\)-dimensional Zakharov system (Q2764647)

The author proves global existence of solutions to the initial value problem for the Zakharov system in one spatial dimension, which reads as NEWLINE\[NEWLINE iu_t+u_{xx}=nu,\quad n_{tt}-n_{xx}=(|u|^2)_{xx},\quad u(0)=u_0,\quad n(0)=n_0, \quad n_t(0)=n_1. NEWLINE\]NEWLINE The main novelty of the paper is that initial data \(u_0\in H^{s, 2}({\mathbb R})\), \(n_0\in L^2({\mathbb R})\), and \(n_1\in \dot{H}^{-1, 2}({\mathbb R})\) are allowed with \({9\over 10}<s<1\). This makes it impossible in the argument to use directly conservation of the energy NEWLINE\[NEWLINEE(t)=\|u_x(t)\|_{L^2}^2+{1\over 2}(\|n(t)\|_{L^2}^2+\|V(t)\|_{L^2}^2) +\int n(t)|u(t)|^2 dx,NEWLINE\]NEWLINE where \(V_x=-n_t\), since this would require solutions (and hence initial data) such that \(u(t)\in H^{1, 2}({\mathbb R})\) for all \(t\). A general approach on how to lower the regularity assumptions on the initial data below the natural energy norm of an equation was introduced by Bourgain in the context of the critical nonlinear Schrödinger equation in two space dimensions [see \textit{J. Bourgain}, ibid. 1998, 253-283 (1998; Zbl 0917.35126)]. NEWLINENEWLINENEWLINEThe main idea is to split the initial data into a low-frequency part which is as regular as desired and a remaining high-frequency part which is small. Then the solution evolving from the low-frequency part is studied, and for this conservation of energy can be utilized to get good bounds. Finally the solution of the original problem is considered to be a small perturbation of the low-frequency solution part. It turns out that bounds from one time interval \([0, T_1]\) to the next time interval \([T_1, 2T_1]\), etc. can be bootstrapped provided that \(T_1\), the frequency cut-off, and the degree of regularity \(s\) are carefully adjusted. NEWLINENEWLINENEWLINEThe main technical tools developed in the paper are new estimates on the nonlinearities in function spaces which are adapted to the dispersive character of the system. The paper is well written and contains full details.