Well-posedness of Cauchy problem for Landau equation in critical Besov space

Abstract

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.