-maximal regularity of nonlocal parabolic equations and applications

  • Xicheng Zhang

    School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, PR China


By using Fourierʼs transform and Fefferman–Steinʼs theorem, we investigate the -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 -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.

Cite this article

Xicheng Zhang, -maximal regularity of nonlocal parabolic equations and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 4, pp. 573–614

DOI 10.1016/J.ANIHPC.2012.10.006