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arXiv · 1612.05676

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

Alin Pogan·Kevin Zumbrun
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Abstract

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 ow 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.

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Alin Pogan, Kevin Zumbrun. 2016-12-16. Center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks. https://arxiv.org/abs/1612.05676

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