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.
Categories: math.AP