Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

Abstract

Abstract. We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.

Global Solutions and Time Decay of the Non-cutoff Vlasov–Maxwell–Boltzmann System in the Whole Space

“…As in [1][2][3]7], we suppose that B(v − u, σ ) is supported on cos θ ≥ 0. Notice also that all the physical parameters have been chosen to be unit for simplicity of presentation.…”

Well-Posedness of Spatially Homogeneous Boltzmann Equation with Full-Range Interaction

“…The more interesting thing is that the estimates obtained including the lower and upper bounds are uniform with respect to the parameter . Finally, we refer to [3][4][5]16,17] for the progress on the well-posedness of the inhomogeneous Boltzmann equation when the initial data are near Maxwellian equilibrium.…”

Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff

“…On pourra trouver plus de détails dans la version anglaise, les preuves complètes faisant l'objet d'une prépublication [4].…”

“…The details of the proof will appear in the full version, [4], and here we only sketch the main steps.…”