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.
@article{AIHPC_2013__30_4_573_0,
author = {Zhang, Xicheng},
title = {$ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
volume = {30},
year = {2013},
pages = {573-614},
doi = {10.1016/j.anihpc.2012.10.006},
mrnumber = {3082477},
zbl = {1288.35152},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_4_573_0}
}
Zhang, Xicheng. $ {L}^{p}$-maximal regularity of nonlocal parabolic equations and applications. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 573-614. doi : 10.1016/j.anihpc.2012.10.006. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_4_573_0/
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