A Transport-Diffusion Kinetic Equation

Raoul Bianchetti; Payam Danesh

doi:10.59973/ipil.360

Authors

Raoul Bianchetti Information Physics Institute, Gosport, Hampshire, United Kingdom - www.informationphysicsinstitute.org
Payam Danesh Department of Biosystems Engineering, Faculty of Agricultural Science, University of Tehran, Tehran, Iran

DOI:

https://doi.org/10.59973/ipil.360

Keywords:

Kinetic Fokker-Planck Equation, Ornstein-Uhlenbeck Operator, Hypocoercivity, Semigroup Well-Posedness, Maxwellian Equilibrium, Fourier-Hermite Discretization

Abstract

This paper studies a linear kinetic equation on a periodic phase space with free transport in position and Ornstein-Uhlenbeck relaxation in velocity. The equation is formulated in the weighted Hilbert space associated with the Maxwellian equilibrium. In that setting, the paper establishes the dissipation identity, conservation of mass, semigroup well-posedness, microscopic coercivity in velocity, and exponential convergence to equilibrium on the zero-mass subspace. The spatially homogeneous problem is treated separately, where the entropy law and exponential entropy decay follow directly from the Gaussian logarithmic Sobolev inequality. A final numerical section presents a Fourier-Hermite discretization and illustrates the same relaxation mechanism at the level of decay curves, spectral localization, and density profiles.

References

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