Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: a semigroup approach in $L^1$-spaces.

In this article we have proved that the solution of the linear transport equation converges in the long time limit to its global equilibrium state at an exponential rate if and only

Theorem 1 ensures that when the time becomes large, the renormalized solutions of the Boltzmann equation (in a bounded domain with boundary conditions (8) or (9)) behave as

The second one ensures that any weak solution of the Landau equa- tion is very smooth under suitable assumptions (in fact, we shall only prove in section 9 that all solutions lie in

Keywords: Boltzmann equation; spatially homogeneous; hard potentials; measure solutions; equilibrium; exponential rate of convergence; eternal solution; global-in-time

Hydrodynamic limit, Boltzmann equation, Hard cutoff poten- tial, Incompressible Navier-Stokes equations, Renormalized solutions, Leray

If we compare the present work to our similar result for the Kac equation, another difficulty is the fact that we treat the case of soft and Coulomb potentials (γ ∈ [−3, 0)) instead

SOBOLEV REGULARIZING EFFECT FOR KAC’S EQUATION In this section, we prove the regularity in weighted Sobolev spaces of the weak solutions for the Cauchy problem of the Kac’s

Villani, On the spatially homogeneous Landau equation for hard potentials, Part I : existence, uniqueness and smothness, Comm.. Partial Differential Equations