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

By Radjesvarane Alexandre, Yoshinori Morimoto, S. Ukai, Chao-Jiang Xu, Tong Yang. 2009

136 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c4a0a7b172c18902461d528b776e3a9c871f4a79

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation (2016, background): “Finally, we highlight the related results of regularisation for the Boltzmann equation without long-range interactions [19, 15, 16], and the related perturbative results for the Landau and (long-range interaction) Boltzmann equation [40, 38, 4, 5, 2, 62, 3].”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “For small perturbations around a Maxwellian, the equation is known to have global smooth solutions [38], [37], [8], [2].”