Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation
Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation
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DOI：
10.2422/2036-2145.201702_001
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发表时间：
2016-07
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影响因子：
--
通讯作者：
François Golse;C. Imbert;C. Mouhot;Alexis F. Vasseur
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文献类型：
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作者：
François Golse;C. Imbert;C. Mouhot;Alexis F. Vasseur
我们将De Giorgi-Nash-Moser理论推广到一类动力学Fokker-Planck方程，并推导出Landau-Coulomb方程的新结果。更准确地说，我们首先研究H{o}lder正则性，并建立了一个Harnack不等式的解决方案，一般的线性方程的系数只是可测的，本质上有界的，即假设没有正则性的系数，以便以后得到结果的非线性问题。这个一般方程具有“II型”次椭圆方程的形式结构，有时也称为Kolmogorov型超抛物方程，但具有粗糙系数：它结合了一阶反对称算子和二阶椭圆算子，该算子仅涉及沿着部分坐标的导数和粗糙系数。然后将这些一般结果应用于具有反幂律的朗道方程$\gamma$\in$ [--d，1]中的非负本质有界弱解，其中质量、能量和熵密度都有界，且质量远离0有界，并推导出这些解的H{o}lder正则性.
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.