Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Abstract

Abstract. For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce existence of classical solutions near equilibrium for the full non-linear models associated, with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof are based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Convergence to Equilibrium for the Linearized Cometary Flow Equation

“…Our main convergence result is proven under assumptions of boundedness and smoothness of solutions, which we are unable to prove. Nevertheless, similar properties have been shown recently for simpler models ( [14], [17]). Assumption 2.2.…”

“…In particular, the following entropy dissipation equality is derived analogously to (17) as a consequence of the symmetry of LQ with respect to ·, · µ :…”

Optimal Estimate of the Spectral Gap for the Degenerate Goldstein-Taylor Model

“…This allows indeed a local control of the dissipative properties of the equation, and hence the solution is locally "attracted" everywhere toward its local equilibrium (see, for instance, [11,7,5,16]). …”

Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations

“…Regarding Sobolev regularity results for perturbative states from equilibrium, a broad work of C. Mouhot and N. Newman [38] was also done around the same period of Herau's and Ben Abdallah-Tayeb's results from [33,2] on existence and regularity associated with the linear Vlasov-Boltzmann equation (1). The authors in [38] study the existence, uniqueness regularity and decay rates for a large general class of linear collisional kinetic models in the torus, including, in particular, the linear collisional integral associated with the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semiclassical linearized fermionic and bosonic relaxation models. More specifically, they showed explicit coercivity estimates on the associated integro-differential operators for some modified Sobolev norms.…”

“…In this case it is easy to find stationary states for relaxation models and their corresponding decay rates to equilibrium [33,38]. We will use these properties to select an appropriate truncation for the computational domain.…”