Global mild solutions of the Landau and non-cutoff Boltzmann equations

Abstract

This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an L ∞x,v framework, similar to that in [49], for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the L ∞ T L 2 v norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables.

Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition

“…Furthermore, the only results we are aware of in the case of long-range interaction, that is, for non-cutoff Boltzmann and Landau collision operators in a bounded domain, are the very recent works of Guo-Hwang-Jang-Ouyang [51] (see also [50]) for the Landau equation with specular reflection boundary condition, and Duan-Liu-Sakamoto-Strain [41] for non-cutoff Boltzmann and Landau equations in a finite channel with inflow or specular reflection boundary conditions. However, as far as we understand, the arguments presented in [51] seem to be constructive only when ∂Ω is flat, while the arguments presented in [50] are again non-constructive.…”

“…We also refer to the works [76,77] that establish decay estimates for the non-cutoff Boltzmann and Landau equations with very soft potentials, as well as [25] for the Landau equation. All these results are established in the torus or the whole space, and, to the best of our knowledge, the only works concerning domains with boundary conditions are the recent results of [51] for the Landau equation with specular reflection boundary condition, and [41] for non-cutoff Boltzmann and Landau equations in a finite channel with specular reflection or inflow boundary conditions. Concerning Fokker-Planck equations and kinetic Fokker-Planck equations we shall quote [73,59] and [23], as well as the references therein.…”