Abstract
This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting.
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This paper is dedicated to Professor Philippe G. Ciarlet on the occasion of his 80th birthday.
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Duan, R., Wang, Y. & Zhang, Z. The Boltzmann Equation with Time-periodic Boundary Temperature. Acta Math. Appl. Sin. Engl. Ser. 35, 174–208 (2019). https://doi.org/10.1007/s10255-019-0803-0
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DOI: https://doi.org/10.1007/s10255-019-0803-0
Keywords
- Boltzmann equation
- time-periodic boundary
- time-periodic solutions
- existence
- dynamical stability
- a priori estimates