Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks |
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Authors: | Alin Pogan Kevin Zumbrun |
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Affiliation: | 1. Miami University, Department of Mathematics, 301 S. Patterson Ave., Oxford, OH 45056, USA;2. Indiana University, Department of Mathematics, 831 E. Third St., Bloomington, IN 47405, USA |
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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 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. |
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Keywords: | Degenerate evolution equation Center manifold Steady Boltzmann equation Boltzmann shock profile Boltzmann boundary layer |
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