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2022, Volume 15, Issue 4: 729-752. Doi: 10.3934/krm.2022012
This issue Previous Article Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach Next Article Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $

Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

  • 1.

    Kent State University, Kent, OH 44240

  • 2.

    Indiana University, Bloomington, IN 47405

  • *Corresponding author: Kevin Zumbrun

    *Corresponding author: Kevin Zumbrun

In memory of Robert T. Glassey
Preface by Kevin Zumbrun: It is a great privilege to submit a paper to this memorial issue in honor of long-time mentor and colleague Bob Glassey. Bob enlivened the department at Indiana University for many years with his gentle wit and incisive mathematical analysis. Bob is perhaps best known for his work on kinetic equations. What I know about Boltzmann's equation I know mainly from sitting in on his delightful graduate course, which later became his delightful text on the Cauchy problem for kinetic equations. So, this paper concerning Boltzmann's and related kinetic equations seems an appropriate submission, coming as a direct result of Bob's influence as a teacher and researcher. Of course, Bob is equally well known for his seminal work on blowup in nonlinear PDEs. So, it is perhaps also appropriate that our central estimate is a "blow-down" result consisting of the reverse-time version of finite-time blowup for the Riccati equation.

Received:  September 2021
Revised:  December 2021
Early access:  April 2022
Published:  August 2022

Research of F.N. was partially supported under NSF grant no. DMS-1600239; research of K.Z. was partially supported under NSF grant no. DMS-0300487

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    Abstract

  • We establish an instantaneous smoothing property for decaying solutions on the half-line $ (0, +\infty) $ of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of $ H^1 $ stable manifolds of such equations, showing that $ L^2_{loc} $ solutions that remain sufficiently small in $ L^\infty $ (i) decay exponentially, and (ii) are $ C^\infty $ for $ t>0 $, hence lie eventually in the $ H^1 $ stable manifold constructed by Pogan and Zumbrun.

    Mathematics Subject Classification: Primary: 35Q35, 35Q20; Secondary: 76P05, 82C40.

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