Velocity Averaging and Hölder Regularity for Kinetic Fokker--Planck Equations with General Transport Operators and Rough Coefficients

Zhu, Yuzhe

doi:10.1137/20m1372147

Cited by 13 publications

(6 citation statements)

References 19 publications

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“…Many articles on KFP equations in divergence form are concerned with the local boundedness, Harnack inequality (including a nonhomogeneous version), and Hölder continuity of solutions to (1.1) (see [22], [28], [9], [19], [10], [11], [29]). See also the references in [2].…”

Section: Related Work Divergence Form Equationsmentioning

confidence: 99%

Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

Dong¹,

Yastrzhembskiy²

2022

Preprint

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We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable v and satisfy a vanishing mean oscillation (VMO) type condition. We consider the L 2 case separately and treat more general equations which include the relativistic KFP equation. This work is a continuation of [8], where the present authors studied the KFP equations in nondivergence form. Contents 1. Introduction and the main results 1.1. Notation and assumptions 1.2. Function spaces. 1.3. Main results 1.4. L 2 theory for the kinetic equations with bounded measurable coefficients 1.5. Motivation 1.6. Related works 1.7. Organization of the paper 2. S p -estimate for the model equation 3. Mixed-norm estimate for the model equation 3.1. Proof of Proposition 3.3 3.2. Proof of Theorem 3.1 4. Proof of Theorem 1.12 4.1. Proof of the assertion (i) 4.2. Proof of Theorem 1.12 (ii) and (iii).

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Section: Related Work Divergence Form Equationsmentioning

confidence: 99%

Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

Dong¹,

Yastrzhembskiy²

2022

Preprint

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“…Such a Harnack inequality implies in particular the strong maximum principle [1] relying on a geometric construction known as Harnack chains. The Hölder regularity result of [39] was extended by Y. Zhu [42] to general transport operators ∂ t +b(v)•∇ v for some non-linear function b.…”

Section: Expansion Of Positivity Ferretti and Safonovmentioning

confidence: 99%

Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations

Guerand¹,

Imbert²

2021

Preprint

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This article deals with kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for non-negative super-solutions is derived by considering their Log-transform and following S. N. Kruzhkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

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“…In particular, the interest has been in the development of estimates in continuity spaces, such as Hölder spaces. A suitable Harnack inequality has been proven [23,29,56,57,60], which yields the Hölder regularity of solutions to divergence form kinetic Fokker-Planck equations when the coefficients are merely bounded and elliptic-in-v (note that (1.1) is in non-divergence form). A Harnack inequality for non-divergence form kinetic operators remains elusive [51].…”

Section: Introductionmentioning

confidence: 99%

Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation

Henderson¹,

Wang²

2022

Preprint

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We prove a Schauder estimate for kinetic Fokker-Planck equations that requires only Hölder regularity in space and velocity but not in time. As an application, we deduce a weakstrong uniqueness result of classical solutions to the spatially inhomogeneous Landau equation beginning from initial data having Hölder regularity in x and only a logarithmic modulus of continuity in v. This replaces an earlier result requiring Hölder continuity in both variables.

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Velocity Averaging and Hölder Regularity for Kinetic Fokker--Planck Equations with General Transport Operators and Rough Coefficients

Cited by 13 publications

References 19 publications

Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

Global $L_p$ estimates for kinetic Kolmogorov-Fokker-Planck equations in divergence form

Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations

Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation

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