$L^p$-maximal regularity of nonlocal parabolic equation and applications

Abstract

By using Fourier's transform and Fefferman-Stein's theorem, we investigate the $L^p$-maximal regularity of nonlocal parabolic and elliptic equations with singular and non-symmetric L\'evy operators, and obtain the unique strong solvability of the corresponding nonlocal parabolic and elliptic equations, where the probabilistic representation plays an important role. In particular, a characterization for the domain of pseudo-differential operators of L\'evy type with singular kernels is given in terms of the Bessel potential spaces. As a byproduct, we show that a large class of non-symmetric L\'evy operators generates an analytic semigroup in $L^p$-space. Moreover, as applications, we prove a Krylov's estimate for stochastic differential equation driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the well-posedness to a class of quasi-linear first order parabolic equation with critical diffusion. In particular, critical Hamilton-Jacobi equation and multidimensional critical Burger's equation are uniquely solvable and the smooth solutions are obtained.