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Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation

Anton Arnold, Gayrat Toshpulatov

Abstract

We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted $L^2$ and Sobolov spaces. Our result holds for a wide class of external potentials and the estimates on the rate of convergence are explicit and constructive. Moreover, we prove that the associated semigroup of the equation has hypoelliptic regularizing properties and we obtain explicit rates on this regularization. The technique is based on the construction of suitable Lyapunov functionals.

Trend to equilibrium and hypoelliptic regularity for the relativistic Fokker-Planck equation

Abstract

We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted and Sobolov spaces. Our result holds for a wide class of external potentials and the estimates on the rate of convergence are explicit and constructive. Moreover, we prove that the associated semigroup of the equation has hypoelliptic regularizing properties and we obtain explicit rates on this regularization. The technique is based on the construction of suitable Lyapunov functionals.
Paper Structure (14 sections, 17 theorems, 256 equations)

This paper contains 14 sections, 17 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Exponential convergence in $L^2(\mathbb{R}^{2d}, f_{\infty})$
  4. The first Lyapunov functional
  5. Proof of Theorem \ref{['main result']}
  6. Exponential convergence in $\mathscr{H}^1(\mathbb{R}^{2d}, f_{\infty})$
  7. Preliminaries
  8. The second Lyapunov functional
  9. Hypoelliptic regularity
  10. Newtonian limit $c\to\infty$
  11. Conclusion and Outlook
  12. Appendix
  13. Weighted Poincaré inequalities and an elliptic regularity result
  14. Some matrix inequalities

Key Result

Theorem 2.2

Let $\frac{f_0}{f_{\infty}}\in L^2(\mathbb{R}^{2d}, f_{\infty})$ and $V$ satisfy Assumption Assumptions i-ii. Then there are explicitly computable constants $C_1>0$ and $\lambda>0$ (independent of $f_0$) such that holds for all $t\geq 0.$

Theorems & Definitions (36)

  • Theorem 2.2: Exponential decay in $L^2(\mathbb{R}^{2d}, f_{\infty})$
  • Theorem 2.3: Exponential decay in $\mathscr{H}^1(\mathbb{R}^{2d}, f_{\infty})$
  • Theorem 2.4: Hypoelliptic regularity from $L^2(\mathbb{R}^{2d}, f_{\infty})$ to $\mathscr{H}^1(\mathbb{R}^{2d}, f_{\infty})$
  • Corollary 2.5
  • Remark 2.6
  • Theorem 3.1: dolbeault2015hypocoercivity
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 26 more