Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 367, Issue 6, Pages 3807-3828Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9947-2015-06012-7
Keywords
Kinetic equations; hypocoercivity; Boltzmann; BGK; relaxation; diffusion limit; nonlinear diffusion; Fokker-Planck; confinement; spectral gap; Poincare inequality; Hardy-Poincare inequality
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Funding
- French-Austrian Amadeus project [13785UA]
- ANR
- Austrian Science Fund [W8]
- European network DEASE
- King Abdullah University of Science and Technology (KAUST) [KUK-I1-007-43]
- ERC grant MATKIT
- Austrian Science Fund (FWF) [W8] Funding Source: Austrian Science Fund (FWF)
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We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted L-2 norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.
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