Abstract
We study the longtime behavior of symmetric solutions of the nonlinear Boltzmann equation and a closely related nonlinear FokkerPlanck equation. If the symmetry of the solutions corresponds to shear flows, the existence of stationary solutions can be ruled out because the energy is not conserved. After anisotropic rescaling both equations conserve the energy. We show that the rescaled Boltzmann equation does not admit stationary densities of Maxwellian type (exponentially decaying). For the rescaled FokkerPlanck equation we demonstrate that all solutions converge to a Maxwellian in the longtime limit, however the convergence rate is only algebraic, not exponential.
Original language  English 

Pages (fromto)  1321–1348 
Number of pages  28 
Journal  SIAM Journal on Mathematical Analysis (SIMA) 
Volume  51 
Issue number  2 
Early online date  18 Apr 2019 
DOIs  
Publication status  Published  31 Dec 2019 
Keywords
 Boltzmann equation
 FokkerPlanck
 Hypocoercivity
 Objective solution
ASJC Scopus subject areas
 Analysis
 Computational Mathematics
 Applied Mathematics
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Karsten Matthies
 Department of Mathematical Sciences  Senior Lecturer
 Probability Laboratory at Bath
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching