Local well-posedness of the Boltzmann equation with polynomially decaying initial data

Christopher Henderson; Stanley Snelson; Andrei Tarfulea

doi:10.3934/krm.2020029

Article Contents

2020, Volume 13, Issue 4: 837-867. Doi: 10.3934/krm.2020029

This issue Previous Article Well-posedness for boundary value problems for coagulation-fragmentation equations Next Article

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

1.
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
2.
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA
3.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received: October 2019

Revised: February 2020

Published: May 2020

The first author was partially supported by NSF grant DMS-2003110. The second author was partially supported by a Ralph E. Powe Award from ORAU. The third author was partially supported by NSF grant DMS-2012333

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Abstract

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $ L^2 $ and $ L^\infty $ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

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Mathematics Subject Classification: Primary: 35Q20, 35A01, 35S05.

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