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.
Keywords
classification of steady states
special macroscopic modes
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
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