Regularity results for nonlocal parabolic equations

By M. Kassmann, Russell W. Schwab. 2013

62 citations recorded by the source corpus.

We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3cc7b1ef9c3fe49dfcadec03e6ed03d82f85681c

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “Actually, there has been lately a substantial literature dealing with nonlocal parabolic equations (see for instance [14, 20, 26] and references therein).”
Regularity theory for general stable operators (2014, background): “1), or related ones, has been widely investigated; see the works by Bass, Kassmann, Schwab, Silvestre, Sztonyk, and Bogdan, among others [1, 23, 22, 3, 39, 21, 2, 37, 5, 6, 7, 9, 24].”
Regularity estimates for elliptic nonlocal operators (2015, background): “urrent work follows the strategy laid out in [20] which, on the one hand, allows the constants to be independent of α for α → 2− and, on the other hand, allows to treat general measures. See [14] and [23] for corresponding results in the parabolic case. ELLIPTIC NONLOCAL OPERATORS 10 The articles [1”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “There is a small survey on the subject in [43].”
Regularity for parabolic integro-differential equations with very irregular kernels (2014, background): “We also point out that the results in this paper and all of the others mentioned (except for [25]), require that the Lévy measure– referred to above as μ(x, dh)– has a nontrivial absolutely continuous part, K dh, with respect to Lebesgue measure (our work allows for a measure with a density plus som”
Quantitative approximation properties for the fractional heat equation (2017, unknown): “For more refined results we refer to for instance [14, 21, 26, 3, 18].”