The optimal decay and propagation of regularity for the inhomogeneous Landau equation in (Lx∞ˆ∩Lxrˆ)Lv2 space

Abstract

We study the Cauchy problem for the inhomogeneous Landau equation with hard potentials and moderately soft potentials in $(\widehat{L_x^{\infty }}\cap \widehat{L_x^r})L_v^2$ space with $1\le r<3$. In perturbation framework, we establish the global existence and optimal time decay rate of the solution with initial datum ${\Vert f_0\Vert }_{(\widehat{L_x^{\infty }}\cap \widehat{L_x^r})L_v^2}+{\Vert {(-\Delta )}^{-\frac{1}{2}}{f}_0\Vert }_{\widehat{L_x^r}{L}_v^2}$ smallness, where ${(-\Delta )}^{-\frac{1}{2}}$ is the Riesz potential, $\widehat{L_x^r}$ is the Fourier-Hertz space equipped with the norm ${\Vert f_0\Vert }_{\widehat{L_x^r}}={\Vert \widehat{f_0}\Vert }_{L_{\xi }^p}$ with $\frac{1}{p}+\frac{1}{r}=1$ and $\widehat{f_0}(\xi ,v)$ is the Fourier transform in space variable. Moreover, we obtain the propagation of Gevrey and Gelfand-Shilov regularity in the space and velocity variables.