SCATTERING AND WELL-POSEDNESS FOR THE ZAKHAROV SYSTEM AT A CRITICAL SPACE IN FOUR AND MORE SPATIAL DIMENSIONS

We study the Cauchy problem for the Zakharov system in spatial dimension d >= 4 with initial datum (u(0), n(0), partial derivative(t)n(0)) is an element of H-k(R-d) x (H) over dot(l) (R-d) x (H) over dot(l-1)(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 := (LtLx2d/(d-2))-L-2, which is related to the endpoint Strichartz estimate.