Regularization estimates and Cauchy theory 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 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.

Mots clés: dissipativity; exponential rate of convergence; Boltzmann equation without cut-off; hard potentials; Cauchy theory; Spectral gap; long-time asymptotic; regularization estimates; pseudodifferential theory

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Hérau, Daniela Tonon, Isabelle Tristani. Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off. 2019. ⟨hal-01599973v3⟩