On the differentiability of the solution to an equation with drift and fractional diffusion

By Luis Silvestre. 2010

112 citations recorded by the source corpus.

We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we prove that the solution becomes immediately differentiable with Holder continuous derivatives. Therefore, the solutions to the equation are classical.

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

View Paper: https://www.semanticscholar.org/paper/b353ce9bb30727402669701df558da4de81a75ad

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions (2019, result): “These results should be compared with the known C and C local estimates for solutions to inhomogeneous fractional Laplace equations by Silvestre, Caffarelli-Stinga, and other authors ([8, 21, 5, 9]).”
On a transport equation with nonlocal drift (2014, methodology): “icular, since we expect this regularizing phenomenon to be still valid for the fractionally dissipative equations, it would imply that the equation (1.4) with γ ≥ 1/2 has global smooth solutions (cf. [41]), thereby answering Conjecture 1 in [28]. We conclude the paper with a related and perhaps weak”
Global well-posedness of slightly supercritical active scalar equations (2012, background): “It was shown by Constantin and Wu [2008] that if the solution is C with δ > 1− α, it is smooth (see also [Silvestre 2011] for drift velocity that is not divergence free).”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, background): “By the regularity theory for fractional Laplacians (see [13,47,63,64,82] and [83]), for any 0 < α < 1, there exists some constant C > 0 which just depends on s, G0 and G1 such that ‖uk‖C2,α(Rn) ≤ C, ∀k ≥ 1.”
Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “[30]) and in the theory of stochastic processes (see e.”
Stochastic flows for Lévy processes with Hölder drifts (2015, background): “However, the approach of [20] to establish (1.”