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

Lukas Niebel, Rico Zacher

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.

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

Abstract

We introduce the concept of kinetic maximal -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 -regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal -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.

Paper Structure

This paper contains 11 sections, 53 theorems, 252 equations.

Table of Contents

  1. Introduction
  2. Kinetic Maximal $L^p_\mu$-regularity with temporal weights
  3. Kinetic Regularization in $L^p_\mu(L^q)$
  4. Kinetic maximal $L^p(L^q)$-regularity for the Kolmogorov equation
  5. Kinetic maximal $L^p(X_\beta^{s,q})$-regularity with temporal weights
  6. Characterization of the trace space of strong solutions to the Kolmogorov equation
  7. Kinetic maximal $L^p_\mu$-regularity for the (fractional) Kolmogorov equation in spaces of higher regularity
  8. The Kolmogorov equation with bounded and uniformly continuous coefficients
  9. Quasilinear Kolmogorov equations
  10. Local existence of strong $L^p_\mu(L^q)$-solutions to a quasilinear kinetic diffusion equation
  11. Appendix

Key Result

Proposition 2.1

For all $p,q \in (1,\infty)$, any $\mu \in (1/p,1]$ and all $T \in (0,\infty]$ the mapping is a well-defined isometric isomorphism. The inverse operator is given by $[\Gamma^{-1}u](t,x,v) = u(t,x-tv,v)$. Moreover, the group $(\Gamma(t))_{t \in \mathbb R} \colon L^q(\mathbb R^{2n}) \to L^q(\mathbb R^{2n})$ of isometries is strongly continuous.

Theorems & Definitions (112)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 102 more