[Paper Review] Kinetic maximal $L^p$-regularity with temporal weights and application to quasilinear kinetic diffusion equations

Lukas Niebel, Rico Zacher|arXiv (Cornell University)|Dec 14, 2020

Numerical methods in inverse problems24 references4 citations

TL;DR

This paper introduces kinetic maximal $L^p$-regularity with temporal weights for the fractional Kolmogorov equation, proving that solutions lie in weighted Lebesgue spaces and are continuous in time with values in anisotropic Besov trace spaces. The key contribution is establishing this regularity for variable-coefficient kinetic equations and applying it to prove local existence of solutions to quasilinear kinetic diffusion equations.

ABSTRACT

We introduce the concept of kinetic maximal $L^p$-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal $L^p_μ$-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal $L^p_μ$-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.

Motivation & Objective

• To develop a new regularity framework—kinetic maximal $L^p$-regularity with temporal weights—for linear and quasilinear kinetic partial differential equations.
• To characterize the trace space of solutions to the kinetic equation in terms of anisotropic Besov spaces, extending classical regularity theory to the kinetic setting.
• To extend the theory of maximal $L^p$-regularity to the Kolmogorov equation with variable coefficients, overcoming the limitation that the standard theory fails due to non-holomorphic semigroups.
• To apply the developed regularity theory to prove local existence of solutions for a class of quasilinear kinetic diffusion equations.
• To establish the instantaneous smoothing property and continuity in time with values in the trace space for solutions to kinetic equations with temporal weights.

Proposed method

• Introduce the space $\mathbb{E}_\mu(0,T) = \mathcal{T}_\mu^p((0,T);L^q(\mathbb{R}^{2n})) \cap L^p_\mu((0,T);H_v^{\beta,q}(\mathbb{R}^{2n}))$ to capture kinetic and fractional diffusion regularity with temporal weights.
• Prove kinetic maximal $L^p_\mu(L^q)$-regularity for the fractional Kolmogorov equation by establishing a weighted $L^p$-estimate for $\partial_t u + v\cdot\nabla_x u$ and $(-\Delta_v)^{\beta/2}u$.
• Use the homogeneous group structure on $\mathbb{R}^{2n+1}$ to analyze the kinetic operator $\partial_t + v\cdot\nabla_x$ and derive estimates invariant under anisotropic scaling.
• Characterize the trace space $X_{\gamma,\mu}$ as anisotropic Besov space $B_{qp}^{\mu-1/p, \alpha}(\mathbb{R}^{2n})$ with $\alpha = (3/2,\dots,3/2,1/2,\dots,1/2)$, linking solution regularity to function space theory.
• Apply Mikhlin multiplier theorem to boundedness of pseudodifferential operators on anisotropic Besov spaces, ensuring continuity of fractional Laplacian and related operators.
• Use a fixed-point argument in the weighted regularity space $\mathbb{E}_\mu(0,T)$ to prove local existence for quasilinear kinetic equations, leveraging the trace space continuity and embedding theorems.

Experimental results

Research questions

• RQ1 Can kinetic maximal $L^p$-regularity with temporal weights be established for the fractional Kolmogorov equation with $p,q \in (1,\infty)$ and $\mu \in (1/p,1]$?
• RQ2 How can the trace space of solutions to kinetic equations be characterized in terms of anisotropic Besov spaces?
• RQ3 Does kinetic maximal $L^p_\mu$-regularity extend to the Kolmogorov equation with variable coefficients?
• RQ4 Can kinetic maximal $L^p_\mu$-regularity be used to prove local existence of solutions to quasilinear kinetic diffusion equations?
• RQ5 What is the role of instantaneous smoothing and continuity in time with values in the trace space for kinetic equations with temporal weights?

Key findings

• Kinetic maximal $L^p_\mu(L^q)$-regularity holds for the fractional Kolmogorov equation with $p,q \in (1,\infty)$ and $\mu \in (1/p,1]$, extending previous results to general $p,q$ and weighted norms.
• Solutions to the kinetic equation are continuous in time with values in the trace space $X_{\gamma,\mu} = (L^q(\mathbb{R}^{2n}), D(A))_{\mu-1/p,p}$, which is identified as the anisotropic Besov space $B_{qp}^{\mu-1/p, \alpha}(\mathbb{R}^{2n})$ with $\alpha = (3/2,\dots,3/2,1/2,\dots,1/2)$.
• The trace space $X_{\gamma,\mu}$ embeds into $C_0(\mathbb{R}^{2n})$ if $\mu - 1/p > 1/2 + 2n/q$, ensuring pointwise continuity of solutions.
• The fractional Laplacian $(-\Delta_v)^{\beta/2}$ maps $B_{qp}^{s,\alpha}(\mathbb{R}^{2n})$ continuously into $B_{qp}^{s-\beta/2,\alpha}(\mathbb{R}^{2n})$, preserving the anisotropic structure.
• For $\mu \in (1/p,1]$, the space ${{}^\mathrm{kin}B}_{qp}^{\mu-1/p,2}(\mathbb{R}^{2n})$ embeds into $C_0^1(\mathbb{R}^{2n})$ if $\mu - 1/p > 1/2 + 2n/q$, implying higher regularity of the solution trace.
• Local existence of solutions to quasilinear kinetic diffusion equations is established via a fixed-point argument in the space $\mathbb{E}_\mu(0,T)$, relying on the kinetic maximal $L^p_\mu$-regularity framework.