Reconstruction of the collision kernel in the nonlinear Boltzmann equation

Ru Yu Lai, Gunther Uhlmann, Yang Yang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We consider an inverse problem for the Boltzmann equation with nonlinear collision operator in dimensions n ≥ 2. We show that the kinetic collision kernel can be uniquely determined from the incoming-to-outgoing mappings on the boundary of the domain provided that the kernel satisfies a monotonicity condition. Furthermore, a reconstruction formula is also derived. The key methodology is based on the higher-order linearization scheme to reduce a nonlinear equation into simpler linear equations by introducing multiple small parameters into the original equation.

Original languageEnglish (US)
Pages (from-to)1049-1069
Number of pages21
JournalSIAM Journal on Mathematical Analysis
Issue number1
StatePublished - 2021

Bibliographical note

Funding Information:
∗Received by the editors April 3, 2020; accepted for publication (in revised form) December 7, 2020; published electronically February 16, 2021. Funding: The work of the first author was supported by National Science Foundation grants DMS-1714490 and DMS-2006731. The work of the second author was partially supported by the National Science Foundation, the Walker Family Endowed Professorship at the University of Washington, and the Si-Yuan Professorship at IAS, HKUST. The work of the third author was partially supported by National Science Foundation grants DMS-1715178 and DMS-2006881 and a startup fund from Michigan State University.

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.


  • Boltzmann equation
  • Collision operator
  • Inverse problems
  • Nonlinearity


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