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arXiv · 1609.02691

Hölder estimates for fractional parabolic equations with critical divergence free drifts

Matias G. Delgadino·Scott Smith
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Abstract

This work focuses on drift-diffusion equations with fractional dissipation $(-Δ)^α$ in the regime $α\in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $β\in (0,1)$, the $C^β$ norm of the solution depends only on the size of the drift in critical spaces of the form $L^{q}_{t}(BMO^{-γ}_{x})$ with $q>2$ and $γ\in (0,2α-1]$, along with the $L^{2}_{x}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.

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Matias G. Delgadino, Scott Smith. 2016-09-09. Hölder estimates for fractional parabolic equations with critical divergence free drifts. https://arxiv.org/abs/1609.02691

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