Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show tha...

Full description

Saved in:
Bibliographic Details
Published in:Journal of Differential Equations 2018-06, Vol.264 (11), p.6752-6808
Main Authors: Pogan, Alin, Zumbrun, Kevin
Format: Article
Language:English
Subjects:
Boltzmann boundary layer
Boltzmann shock profile
Center manifold
Degenerate evolution equation
Steady Boltzmann equation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman–Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2018.01.049