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Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off

Hérau, Frédéric; Tonon, Daniela; Tristani, Isabelle (2020), Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, Communications in Mathematical Physics, 377, 1, p. 697-771. 10.1007/s00220-020-03682-8

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Type
Article accepté pour publication ou publié
Date
2020
Journal name
Communications in Mathematical Physics
Volume
377
Number
1
Publisher
Springer
Pages
697-771
Publication identifier
10.1007/s00220-020-03682-8
Metadata
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Author(s)
Hérau, Frédéric cc
Laboratoire de Mathématiques Jean Leray [LMJL]
Tonon, Daniela
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Tristani, Isabelle
Département de Mathématiques et Applications - ENS Paris [DMA]
Abstract (EN)
In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant framework for this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.
Subjects / Keywords
dissipativity; exponential rate of convergence; Boltzmann equation without cut-off; hard potentials; Cauchy theory; Spectral gap; long-time asymptotic; regularization estimates; pseudodifferential theory

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