Abstract
Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
Highlights
Among the extensive studies on the Boltzmann equation, most of them are based on the Grad’s cutoff assumption to avoid the mathematical difficulty from the grazing effects in the elastic collisions between particles
A lot of progress has made on the study on the non-cutoff problems, cf. [1, 2, 6, 7, 12] and references therein, which shows that the singularity of collision cross-section yields some gain of regularity on weak solutions
This paper is concerned with the smoothing effects of the singular integral kenerl in the collision operator coming from the non-cutoff cross-sections in the Boltzmann equation
Summary
Among the extensive studies on the Boltzmann equation, most of them are based on the Grad’s cutoff assumption to avoid the mathematical difficulty from the grazing effects in the elastic collisions between particles. A lot of progress has made on the study on the non-cutoff problems, cf [1, 2, 6, 7, 12] and references therein, which shows that the singularity of collision cross-section yields some gain of regularity on weak solutions In some sense, this gives the hypo-ellipiticity property of the Boltzmann operator without angular cutoff. Even though the assumption (1.8) on the cross-section is mathematical because the exact cross-section depends on the relative velocity as given in (2.2), the following analysis reveals the smoothing effect of the singularity of the collision operator on the weak solution to the Boltzmann equation. Remark 1.2: (1) L2l (R3) ⊂ L12(R3) when l > 7/2. (2) Though the above results are given when the space dimension equals to three, they hold for any space dimensions with due modification
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