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

2011/09/05 by Xicheng Zhang, Zhang, Xicheng
Mathematics · #45K05 #47G20 #60H30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #math.AP #math.PR #msc:45K05 #msc:47G20 #msc:60H30

paper · pdf · doi:10.48550/arxiv.1109.0816

38 pages, Theorem 6.1 is improved

arxiv created 2012/01/02 · arxiv updated 2012/01/04

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 Lévy 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 Lévy 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 Lévy 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.

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