Special macroscopic modes and hypocoercivity
Résumé
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.
Mots clés
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
hypocoercivity
linear kinetic equations
collision operator
transport operator
collision invariant
local conservation laws
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