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

Abstract

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 frameworkfor 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.

arXiv admin note: text overlap with arXiv:1709.09943

References in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Cauchy problem and exponential stability for the inhomogeneous Landau equation
• Decay estimates for large velocities in the Boltzmann equation without cutoff

Cited by in corpus (6)

• Non-Equilibrium Dynamics and Weakly Broken Integrability
• Regularity of the Vlasov-Poisson-Boltzmann System without angular cutoff
• Non-cutoff Boltzmann equation with soft potentials in the whole space
• On the convergence from Boltzmann to Navier-Stokes-Fourier for general initial data
• Global Existence of Non-cutoff Boltzmann Equation in Weighted Sobolev Space
• Hypoelliptic and spectral estimates for the linearized Landau operator