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

2015/12/02 by Isao Kato, Kato, Isao, Kotaro Tsugawa +1
Mathematics · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1512.00551

Abstract

We study the Cauchy problem for the Zakharov system in spatial dimension d≥ 4 with initial datum (u(0), n(0), ∂_t n(0)) ∈ H^k(ℝ^d) × H^l(ℝ^d)× H^l-1(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 U², V² type spaces introduced by Koch-Tataru. To avoid the difficulty, we use an intersection space of V² type space and the space-time Lebesgue space L²_tL_x^2d/(d-2), which is related to the endpoint Strichartz estimate.

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