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Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation

Lukas Niebel, Rico Zacher

Abstract

We introduce the notion of kinetic maximal $L^2$-regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the regularity of solutions to the fractional Kolmogorov equation in terms of fractional anisotropic Sobolev spaces. It is shown that solutions of the homogeneous (fractional) Kolmogorov equation define a semi-flow in a suitable function space and the property of instantaneous regularization is investigated.

Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation

Abstract

We introduce the notion of kinetic maximal -regularity with temporal weights for the (fractional) Kolmogorov equation. In particular, we determine the function spaces for the inhomogeneity and the initial value which characterize the regularity of solutions to the fractional Kolmogorov equation in terms of fractional anisotropic Sobolev spaces. It is shown that solutions of the homogeneous (fractional) Kolmogorov equation define a semi-flow in a suitable function space and the property of instantaneous regularization is investigated.

Paper Structure

This paper contains 6 sections, 23 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimates in the nonhomogeneous case with vanishing initial data
  4. Estimates in the homogeneous case with nonvanishing initial data
  5. Kinetic maximal $L^2_\mu(X_\beta^s)$-regularity
  6. Instantaneous Regularization of solutions to the homogeneous Kolmogorov equation

Key Result

Theorem 1.1

Let $T>0$ and $\beta \in (0,2]$. The Kolmogorov equation possesses a unique solution $u \in L^2((0,T);L^2(\mathbb R^{2n}))$ satisfying and $u \in C([0,T];H_x^{\frac{\beta/2}{\beta+1}}(\mathbb R^{2n}) \cap H_v^{\beta/2}(\mathbb R^{2n})$ if and only if

Theorems & Definitions (51)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 41 more