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

TL;DR

This paper establishes scattering and global well-posedness for the Zakharov system at the critical space in four or more dimensions, using novel intersection space techniques related to endpoint Strichartz estimates.

Contribution

It introduces a new approach employing an intersection space of $V^2$ type and Lebesgue spaces to handle bilinear estimates at the critical regularity.

Findings

Proves small data global well-posedness at critical space.

Establishes scattering for the Zakharov system in high dimensions.

Develops a new analytical method avoiding traditional bilinear estimate difficulties.

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 $L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.