L p -maximal regularity of nonlocal parabolic equations and applications
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, p. 573-614
By using Fourierʼs transform and Fefferman–Steinʼs 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. 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 L p -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.
DOI : https://doi.org/10.1016/j.anihpc.2012.10.006
Keywords: L p -regularity, Lévy process, Krylovʼs estimate, Sharp function, Critical Burgerʼs equation
@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},
     publisher = {Elsevier},
     volume = {30},
     number = {4},
     year = {2013},
     pages = {573-614},
     doi = {10.1016/j.anihpc.2012.10.006},
     zbl = {1288.35152},
     mrnumber = {3082477},
     language = {en},
     url = {http://www.numdam.org/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, Volume 30 (2013) no. 4, pp. 573-614. doi : 10.1016/j.anihpc.2012.10.006. http://www.numdam.org/item/AIHPC_2013__30_4_573_0/

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