Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions

Abstract

We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $(u(0),n(0),{\partial }_{t}n(0))\in H^k({\mathbb{R}}^d)\times {\dot{H}}^l({\mathbb{R}}^d)\times {\dot{H}}^{l-1}({\mathbb{R}}^d)$. According to Ginibre, Tsutsumi and Velo, the critical exponent of $(k,l)$ is $((d-3)/2,(d-4)/2)$. We prove the scattering and the small data global well-posedness at the critical space. It seems difficult to get the crucial bilinear estimate only by applying the $U^2,V^2$ type spaces introduced by Koch-Tataru. To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $L_t^2{L}_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.

References in corpus (3)

• Well-posedness and scattering for the KP-II equation in a critical space
• Convolutions of singular measures and applications to the Zakharov system
• Local ill-posedness of the 1D Zakharov system