Global well-posedness and scattering for the Zakharov system at the critical space in three spatial dimensions with small and radial initial data

Abstract

We consider the Cauchy problem for the Zakharov system at the critical space in three spatial dimensions. Here, the initial data are small and spherically symmetric. When k and l are the regularity of the initial data of the Schrödinger equation and the wave equation, the critical exponent of the three-dimensional Zakharov system is (k,l)=(0,−12). If the initial data are spherically symmetric, we can show the small data global well-posedness and the scattering in the space with the critical exponent by applying the radial Strichartz estimates.