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 ([9]), the critical exponent of $(k,l)$ is $ ((d-3)/2,(d-4)/2 ). $ We prove the small data global well-posedness and the scattering 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 and Tataru ([23], [24]). To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.