Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

Alin Pogan; Kevin Zumbrun

doi:10.3934/krm.2019001

Article Contents

2019, Volume 12, Issue 1: 1-36. Doi: 10.3934/krm.2019001

This issue Previous Article Next Article Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density

Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

Alin Pogan^1, and
Kevin Zumbrun^2, ,

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

^* Corresponding author: Kevin Zumbrun
^* Corresponding author: Kevin Zumbrun

Received: April 30, 2017

Published: July 2018

Research of A. P. was partially supported under the Summer Research Grant program, Miami University and by the Simons Foundation. Research of K.Z. was partially supported under NSF grant no. DMS-0300487.

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Abstract

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

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Mathematics Subject Classification: Primary: 47J06; Secondary: 47J35, 76P05.

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