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Special macroscopic modes and hypocoercivity

Carrapatoso, Kleber; Dolbeault, Jean; Herau, Frederic; Mischler, Stéphane; Mouhot, Clément; Schmeiser, Christian (2022), Special macroscopic modes and hypocoercivity. https://basepub.dauphine.psl.eu/handle/123456789/23707

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
41
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Author(s)
Carrapatoso, Kleber
Centre de Mathématiques Laurent Schwartz [CMLS]
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Herau, Frederic cc
Laboratoire de Mathématiques Jean Leray [LMJL]
Mischler, Stéphane
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Mouhot, Clément
Department of Pure Mathematics and Mathematical Statistics [DPMMS]
Schmeiser, Christian
Abstract (EN)
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.
Subjects / Keywords
hypocoercivity; linear kinetic equations; collision operator; transport operator; collision invariant; local conservation laws; micro/macro decomposition; commutators; hypoellipticity; confinement potential; rotations; rotational invariance; symmetries; global conservation laws; spectral gap; Poincaré-Korn inequality; Witten-Laplace operator; partially harmonic potential; time-periodic solutions; classification of steady states; special macroscopic modes