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Local Well-Posedness for Boltzmann’s Equation and the Boltzmann Hierarchy via Wigner Transform

  • Thomas Chen
  • Ryan Denlinger
  • Nataša PavlovićEmail author
Article
  • 12 Downloads

Abstract

We use the dispersive properties of the linear Schrödinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain \({\mathbb{R}^d}\) for \({d\geq 2}\) . The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data f0 are weighted versions of the Sobolev spaces \({L^2_v H^\alpha_x}\) with \({\alpha \in \left( \frac{d-1}{2},\infty\right)}\) . Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell molecules (but not hard spheres); the latter result holds without any factorization assumption for the initial data.

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Notes

Acknowledgements

The work of T.C. is supported by NSF Grants DMS-1151414 (CAREER),DMS-1262411, and DMS-1716198. R. D. gratefully acknowledges support from a postdoctoral fellowship at the University of Texas at Austin. The work of N.P. is supported by NSF Grant DMS-1516228.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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