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

Yuzhe Zhu|arXiv (Cornell University)|Oct 8, 2020

Gas Dynamics and Kinetic Theory23 references4 citations

TL;DR

This paper establishes local boundedness and Hölder regularity for weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients by combining velocity averaging lemmas with De Giorgi-Nash-Moser theory. The key contribution is proving these regularity properties under minimal nondegeneracy conditions on the transport field $ b(v) $, extending prior results from the $ b(v) = v $ case to general $ b \in L^\infty \cap H^1 $ with controlled level sets.

ABSTRACT

This article addresses the local boundedness and Hölder continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and transport. Although the equation is parabolic only in the velocity variable, it has a hypoelliptic structure provided that the transport part $\partial_t+b(v)\cdot abla_x$ is nondegenerate in some sense. We achieve the results by revisiting the method, proposed by Golse, Imbert, Mouhot and Vasseur in the case $b(v)= v$, that combines the elliptic De Giorgi-Nash-Moser theory with velocity averaging lemmas.

Motivation & Objective

• To establish local boundedness of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients.
• To prove Hölder continuity of solutions under a nondegeneracy condition on the transport field $ b(v) $.
• To extend the De Giorgi-Nash-Moser theory to kinetic equations with general $ b(v) $, not just $ b(v) = v $.
• To show that velocity averaging lemmas can compensate for lack of elliptic regularity in the velocity variable when $ b(v) $ is nondegenerate.
• To derive quantitative estimates on the oscillation and $ C^\beta $ norm of solutions in terms of data norms.

Proposed method

• Revisits the method of Golse, Imbert, Mouhot, and Vasseur, combining velocity averaging lemmas with De Giorgi-Nash-Moser theory for kinetic equations.
• Uses a nondegeneracy condition on $ b(v) $, expressed as a level set measure bound: $ \big{|}\{v \in B_r(v_0) : |\mu + b(v)\cdot\nu| \leq \epsilon\}\big{|} \leq K r^{d_2-1} \epsilon $, to control the mixing effect of transport and diffusion.
• Applies a rescaling argument to strengthen the nondegeneracy assumption for Hölder regularity, requiring $ b \in C^1 $ with controlled modulus of continuity of $ Db $.
• Employs a modified De Giorgi iteration scheme in a kinetic geometry, using intrinsic parabolic cylinders $ \mathcal{C}_r $ to handle the hypoelliptic structure.
• Introduces a cutoff and truncation technique to control the positive and negative parts of the solution, reducing the problem to subsolution estimates.
• Uses a qualitative isoperimetric lemma to show that if the derivative of a characteristic function is bounded in $ L^p $, then it must be nonpositive, which helps control oscillation.

Experimental results

Research questions

• RQ1 Can local boundedness be established for kinetic Fokker-Planck equations with general transport fields $ b(v) $ and rough coefficients?
• RQ2 Under what conditions on $ b(v) $ does the solution to the kinetic Fokker-Planck equation achieve Hölder continuity?
• RQ3 To what extent can velocity averaging lemmas compensate for the lack of ellipticity in the velocity variable?
• RQ4 How does the nondegeneracy of $ b(v) $, measured via level set thickness, affect the regularity of solutions?
• RQ5 Can the De Giorgi-Nash-Moser theory be adapted to kinetic equations with non-affine transport operators?

Key findings

• Local boundedness holds for subsolutions under the nondegeneracy condition (1.3), with $ f^+ $ bounded in $ Q_{\text{int}} $ provided $ s \in L^q $ for $ q > \frac{(4+\alpha)(4+d_2)(1+d_1+d_2)}{2\alpha} $.
• The $ L^\infty $ norm of $ f^+ $ in $ Q_{\text{int}} $ is controlled by $ \|f^+\|_{L^2(Q_1)} + \|s\|_{L^q(Q_1)} $, with a constant depending only on $ \lambda, \Lambda, d_1, d_2, K, \alpha, q $.
• Hölder regularity is established for weak solutions when $ d_1 = d_2 = d $, $ b \in C^1 $, and the nondegeneracy condition (1.4) holds with $ b \in C^1 $ and $ Db $ having modulus of continuity $ o_1 $.
• The solution satisfies $ \|f\|_{C^\beta(Q_{\text{int}})} \leq C(\|f\|_{L^2(Q_1)} + \|s\|_{L^q(Q_1)}) $ for $ q > (1+2d)^2 $, with $ \beta \in (0,1) $ and $ C $ depending on $ \lambda, \Lambda, d, K, q, o_1 $.
• The oscillation of the solution in a smaller cylinder decays geometrically: $ \text{osc}_{\mathcal{C}_\omega} \tilde{f} \leq (1-\theta_0)\text{osc}_{\mathcal{C}_1} \tilde{f} + C\|s\|_{L^q} + \text{osc}_{B_1 \times B_1} f(0,\cdot,\cdot) $, leading to Hölder continuity.
• A quantitative Hölder estimate is derived: $ \text{osc}_{B_{cr^3}(z_0)} f \lesssim r^{3\beta_0}(\|f\|_{L^2} + \|s\|_{L^q} + \|f(0,\cdot,\cdot)\|_{C^{\alpha_0}}) $ with $ \beta_0 = \frac{1}{3}\min(\frac{\alpha_0}{2}, \beta_1) $.