Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions. (English) Zbl 1424.35314

Summary: We study the Cauchy problem for the Zakharov system in spatial dimension \(d\geq 4\) 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 J. Ginibre, Y. Tsutsumi and G. Velo [J. Funct. Anal. 151, No. 2, 384–436 (1997; Zbl 0894.35108)], 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 H. Koch and D. Tataru [Commun. Pure Appl. Math. 58, No. 2, 217–284 (2005; Zbl 1078.35143); Int. Math. Res. Not. 2007, No. 16, Article ID rnm053, 36 p. (2007; Zbl 1169.35055)]. To avoid the difficulty, we use an intersection space of \(V^2\) type space and the space-time Lebesgue space \(E:=L_t^2L_x^{2d/(d-2)}\), which is related to the endpoint Strichartz estimate.


35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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