Hölder estimates for fractional parabolic equations with critical divergence free drifts

Delgadino, Matías G.; Smith, Scott

doi:10.1016/j.anihpc.2017.06.004

Delgadino, Matías G. ; Smith, Scott

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 577-604.

Résumé

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 $L_{t}^{q} ({BMO}_{x}^{- γ})$ with $q > 2$ and $γ \in (0, 2 α - 1]$ , along with the $L_{x}^{2}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.

MR Zbl

DOI : 10.1016/j.anihpc.2017.06.004

Mots clés : Fractional parabolic equations, BMO, De Giorgi's method for non-local equations, Drift in critical spaces, Regularity

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     author = {Delgadino, Mat{\'\i}as G. and Smith, Scott},
     title = {H\"older estimates for fractional parabolic equations with critical divergence free drifts},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {577--604},
     publisher = {Elsevier},
     volume = {35},
     number = {3},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.06.004},
     mrnumber = {3778643},
     zbl = {1391.35083},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2017.06.004/}
}

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Delgadino, Matías G.; Smith, Scott. Hölder estimates for fractional parabolic equations with critical divergence free drifts. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 3, pp. 577-604. doi : 10.1016/j.anihpc.2017.06.004. http://www.numdam.org/articles/10.1016/j.anihpc.2017.06.004/

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