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.
Bibliographical noteFunding Information:
∗Received by the editors April 3, 2020; accepted for publication (in revised form) December 7, 2020; published electronically February 16, 2021. https://doi.org/10.1137/20M1329366 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.
© 2021 Society for Industrial and Applied Mathematics.
- Boltzmann equation
- Collision operator
- Inverse problems