An Inverse Problem for the Relativistic Boltzmann Equation
Balehowsky, T., Kujanpää, A., Lassas, M., & Liimatainen, T. (2022). An Inverse Problem for the Relativistic Boltzmann Equation. Communications in Mathematical Physics, 396(3), 9831049. https://doi.org/10.1007/s00220022044868
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Communications in Mathematical PhysicsDate
2022Discipline
Inversioongelmien huippuyksikköMatematiikkaCentre of Excellence in Inverse ProblemsMathematicsCopyright
© 2022 Springer
We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood V⊂M of a timelike path μ that connects a point x− to a point x+. The measurements are modelled by a sourcetosolution map, which maps a source supported in V to the restriction of the solution to the Boltzmann equation to the set V. We show that the sourcetosolution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set
I+(x−)∩I−(x+)⊂M. The set I+(x−)∩I−(x+) is the intersection of the future of the point x− and the past of the point x+, and hence is the maximal set to where causal signals sent from x− can propagate and return to the point x+. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.
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https://converis.jyu.fi/converis/portal/detail/Publication/156520596
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Research Council of FinlandFunding program(s)
Academy Project, AoF; Centre of Excellence, AoFAdditional information about funding
The authors were supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Modelling and Imaging, Grant numbers 312121 and 309963) and AtMath Collaboration project.License
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