Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off
Tristani, Isabelle (2014), Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off, Journal of Statistical Physics, 157, 3, p. 474-496. http://dx.doi.org/10.1007/s10955-014-1066-z
TypeArticle accepté pour publication ou publié
Journal nameJournal of Statistical Physics
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off with a moderate angular singularity and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani (Commun Math Phys 234(3): 455–490, 2003) where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a L1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. (http://hal.archives-ouvertes.fr/ccsd-00495786, 2013). We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.
Subjects / Keywordsdissipativity; Boltzmann equation without cut-off; long-time asymptotic; exponential rate of convergence; hard potentials; spectral gap
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