Affordable Access

Publisher Website

[formula omitted]-maximal regularity of nonlocal parabolic equations and applications

Authors
Journal
Annales de l Institut Henri Poincare (C) Non Linear Analysis
0294-1449
Publisher
Elsevier
Volume
30
Issue
4
Identifiers
DOI: 10.1016/j.anihpc.2012.10.006
Keywords
  • [Formula Omitted]-Regularity
  • Lévy Process
  • KrylovʼS Estimate
  • Sharp Function
  • Critical BurgerʼS Equation
Disciplines
  • Mathematics

Abstract

Abstract By using Fourierʼs transform and Fefferman–Steinʼs theorem, we investigate the Lp-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. As a consequence, 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 also show that a large class of non-symmetric Lévy operators generates an analytic semigroup in Lp-spaces. Moreover, as applications, we prove Krylovʼs estimate for stochastic differential equations driven by Cauchy processes (i.e. critical diffusion processes), and also obtain the global well-posedness for a class of quasi-linear first order parabolic systems with critical diffusions. In particular, critical Hamilton–Jacobi equations and multidimensional critical Burgerʼs equations are uniquely solvable and the smooth solutions are obtained.

There are no comments yet on this publication. Be the first to share your thoughts.