Regularity for solutions of non local parabolic equations

By Hector A. Chang Lara, Gonzalo Dávila. 2011

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We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We prove Cα regularity in space and time and, under different assumptions on the kernels, C1,α in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985) and Wang (Commun. Pure Appl. Math. 45(1), 27–76, 1992). Our results remain uniform as σ → 2 allowing us to recover most of the regularity results found in Tso (Commun. Partial Diff. Equ. 10(5):543–553, 1985).

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/7046f2441dbbf160931d95ed6d25b8d3ecacfeae

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Optimal results for the fractional heat equation involving the Hardy potential (2014, methodology): “−u(x0,t0 −h) h + Z RN u(x0,t0)−u(y,t0) |x−y|N+2s dy= f(x,t) Therefore, uis a viscosity subsolution. Analogously, we prove that uis a viscosity supersolution. Then we can use the regularity results in [16, 17, 32]. In particular, by using Corollaries 2.6. 2.7 in [32], for a second member Ho¨lder cont”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “There are also many important results in this direction including [17], [60], [16], [15], [57], [22], [58], [13], [12], [19], [23], [25], [47], [26], [42] and [56].”
Schauder estimates for solutions of linear parabolic integro-differential equations (2014, background): “6 in [9] such that, rough speaking, if u is a test function at (x, t), theIu is continuous near (x, t).”
Regularity results for nonlocal parabolic equations (2013, methodology): “1) was treated by Chang Lara and Dávila in [15], where they prove robust and local Hölder regularity via a hybrid of the methods from the stationary case [11] and a modification of the techniques of [48].”
Local Regularity for Parabolic Nonlocal Operators (2012, background): “[8, 13–15] and recently for fully nonlinear nonlocal parabolic equations in [9].”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “k) Harnack inequalities, the Ho¨lder continuity of the solutions has been proved in [21, 27] for instance. For the case of fully nonlinear (i.e. in non-divergence form) nonlocal equations we refer to [15, 16] for Ho¨lder continuity results. In the case of (possibly degenerate/singular)nonlineardiﬀus”
Integro-Differential equations with nonlinear directional dependence (2014, background): “This was first presented in [9], and has been successfully modified for use in the non-symmetric as well as the parabolic settings ([13], [14], [15] ).”