Cauchy theory and exponential stability for inhomogeneous boltzmann equation for hard potentials without cut-off

Frédéric Hérau (1) , Daniela Tonon (2) , Isabelle Tristani (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 DMA - Département de Mathématiques et Applications - ENS Paris

In this paper, we investigate both the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cutoff. 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. Mathematics Subject Classification (2010): 76P05 Rarefied gas flows, Boltzmann equation; 47H20 Semigroups of nonlinear operators; 35B40 Asymptotic behavior of solutions .

Mots clés: dissipativity; exponential rate of convergence; Boltzmann equation without cut-off; hard potentials; Cauchy theory; spec- tral gap; long-time asymptotic

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Hérau, Daniela Tonon, Isabelle Tristani. Cauchy theory and exponential stability for inhomogeneous boltzmann equation for hard potentials without cut-off. 2018. ⟨hal-01599973v2⟩