Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

By François Golse, C. Imbert, Clément Mouhot, Alexis Vasseur. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 2016

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We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{o}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law $\gamma$ $\in$ [--d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the H{o}lder regularity of these solutions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5a9834efd3ce3f9ce5589a74d025f1e6d1632ec

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Citations in the corpus

A survey on the classical theory for Kolmogorov equation (2019, background): “• Harnack inequality: Golse, Imbert, Mouhot and Vasseur prove the Hölder continuity and a Harnack inequality for weak solutions to the kinetic Kolmogorov-Fokker-Planck equation (see [37]).”
The Schauder estimate for kinetic integral equations (2018, background): “The results in [6] and [10] are oblivious of the choice of distance.”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, methodology): “We do not use either averaging lemmas, or hypoelliptic estimates for the x variable (like in [36] or [55]).”
Enhanced dissipation and Hörmander's hypoellipticity (2021, background): “Golse, Imbert, Mouhot, and Vasseur [26] studied the Harnack inequality for kinetic Fokker-Planck and Landau equations.”