Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients

Zhu, Yuzhe

doi:10.48550/arxiv.2010.03867

Cited by 3 publications

(4 citation statements)

References 28 publications

(39 reference statements)

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“…We remark that, armed with Lemma 4.1, the assertions (i) and (ii) in the above lemma directly follow from [20,Proposition 4.4] and [37,Corollary 4.6], respectively. Proposition 4.8 (Existence).…”

Section: Lemma 47 (Hölder Estimates)mentioning

confidence: 80%

“…in Ω, by Lemma 4.1 and the fact that R [w] ≥ ε. In particular, the lower order term R [w] d 2 − |v| 2 4 g 1 for any w ∈ K. Thus, the global Hölder estimate [37,Corollary 4.6] implies that there exist some constants γ ∈ (0, 1) and N > 0 depending only on universal constants and ε such that g C 2γ l (Ω) ≤ N. It then follows from Proposition 3.3 with the interior Schauder estimate (Proposition 3.1) that the mapping…”

Section: Lemma 47 (Hölder Estimates)mentioning

confidence: 98%

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On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

Anceschi¹,

Zhu²

2021

Preprint

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We investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. Its solution describes the density evolution of interacting particles whose mobility is hampered by their aggregation. When the initial data lies below a Maxwellian, we derive the global wellposedness with instantaneous smoothness. The proof relies on hypoelliptic analogue of the classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global in time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy and barrier function methods. Contents1. Introduction 1 2. Preliminaries 6 3. Kolmogorov-Fokker-Planck equation 8 4. Well-posedness of the nonlinear model 11 5. Diffusion asymptotics 22 Appendix A. Maximum principle 30 Appendix B. Spreading of positivity 31 Appendix C. Gaining regularity of spatial increment 33 References 34

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Section: Lemma 47 (Hölder Estimates)mentioning

confidence: 80%

Section: Lemma 47 (Hölder Estimates)mentioning

confidence: 98%

On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

Anceschi¹,

Zhu²

2021

Preprint

Self Cite

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“…We emphasize that all results mentioned concern linear equations. Zhu [41] proved local boundedness and local Hölder continuity of weak solutions of (1.6) when the drift term…”

Section: Theorem 15 (Existence and Uniquenessmentioning

confidence: 99%

On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

Garain¹,

Nyström²

2022

Preprint

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We consider nonlinear Kolmogorov-Fokker-Planck type equations of the formThe functionassumed to be continuous with respect to ξ, and measurable with respect to X, Y and t. A = A(ξ, X, Y, t) is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded X, Y and t dependent domains.

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“…The combination of these ideas saw a lot of recent interest [2,3,10,12,13,14,29,30,31] as it is a path for regularity results for nonlinear kinetic equations, where the solution satisfies schematically…”

Section: Introduction 1motivationmentioning

confidence: 99%

Regularity for rough hypoelliptic equations

Dietert¹,

Hirsch²

2022

Preprint

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Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients

Cited by 3 publications

References 28 publications

On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics

On regularity and existence of weak solutions to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients

Regularity for rough hypoelliptic equations

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