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Special macroscopic modes and hypocoercivity

Kleber Carrapatoso, Jean Dolbeault, Frédéric Hérau, Stéphane Mischler, Clément Mouhot, Christian Schmeiser

Abstract

We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

Special macroscopic modes and hypocoercivity

Abstract

We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator admitting several local conservation laws (local density, momentum and energy). We classify all special macroscopic modes (stationary solutions and time-periodic solutions). We also prove the convergence of all solutions of the evolution equation to such non-trivial modes, with a quantitative exponential rate. This is the first hypocoercivity result with multiple special macroscopic modes with constructive estimates depending on the geometry of the potential.

Paper Structure

This paper contains 52 sections, 39 theorems, 383 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equation and assumptions
  3. The main result
  4. Framework, comments and methods
  5. Conservation laws and macroscopic equations
  6. The equations for the special macroscopic modes
  7. Global conservation laws
  8. A micro-macro decomposition
  9. The equations for the macroscopic modes
  10. Classification of the special macroscopic modes
  11. Statement and preliminary results
  12. Control of A
  13. Control of b, b", c' and c"'
  14. Control of b', c" and c
  15. A Lyapunov function method
  16. ...and 37 more sections

Key Result

Theorem 1.1

Assume that the potential $\phi$ and the collision operator $\mathscr{C}$ satisfy the assumptions eq:kersC--eq:hyp-sg-C--eq:lbound--hyp:intnorm--hyp:regularity--eq:poincarenormal--eq:momentspace--eq:phiid--hyp:semigroup. Then

Figures (1)

  • Figure 1: Complex spectrum of $-\mathscr{L}$.

Theorems & Definitions (76)

  • Theorem 1.1: Special macroscopic modes and hypocoercivity
  • Proposition 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:orthn']}
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 3.1: Special macroscopic modes
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:piL2']}
  • Lemma 3.3
  • ...and 66 more