Some research about the regularity of the solutions for spatially homogeneous Boltzmann equation without angular cutoff
一些關於無截斷假設的空間齊次 Boltzmann 方程解的正則性研究
Student thesis: Doctoral Thesis
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Award date  15 Jul 2010 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(05f502c24666424a915ffab97f09ede8).html 

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Abstract
As one of the most important problems in physicists and mathematicians, the regularity
research for the solution of the Boltzmann equation, has attracted many people
in recent years, its physical significance is to show the smoothness property of the
oneparticle distribution function for the simple monatomic gas. In the early study of
this field, to avoid the mathematical difficulty which from the grazing effects in elastic
collisions between particles, most of the works on the Boltzmann equation are based
on Grad’s cutoff assumption, among them, a research result shows that in weighted Lp
space, the solution preserves the same regularity as the initial data. On the other hand,
it is know that if we remove this assumption, the Boltzmann collision operator behaves
like a fractional Laplacian and hence the Boltzmann equation has been expected to
exhibit the smoothing effects on the solutions.
To the Sobolev regularity of the solutions for the homogeneous Boltzmann equation,
so far there have many advanced research results. It has been justified satisfactorily
for the spatially homogeneous case. Among them, we will introduce our work
in Chapter 2 of this thesis, that is, for the nonMaxwellian molecules case with the
DebyeYukawa potential, under the condition of the Lipschitz continuity, the positive
weak solution of the spatially homogeneous Boltzmann equation lies in the Sobolev
space H+1
loc (R3).
Furthermore, in order to gain the higher order regularity, Chapter 3 is devoted
to study the Gevrey smoothing property for the solution of the spatially homogeneous
Boltzmann equation with the inverse power law potential model. In the first section, we
do a simple introduction on this issue and list some results for the Boltzmann equation
and other related equations in recent years. And then in the second section, we consider the corresponding linearized Cauchy problem in the nonMaxwellian molecules case,
it is worth to mention that the Maxwellian molecules case has been solved in [25]. We
will give a new method to get the Gevrey regularity of the solutions in the local space,
and still without any Gevrey regularity assumption for the initial data. Compare with
the method in [25], the method is base on mathematical induction, including not only
the abstract pseudodifferential calculus, but also many other careful and advanced
mathematical analysis techniques such as the Cauchy integral theorem, etc. More importantly,
it is suitable for the more complex situation such as the nonMaxwellian
molecules case, which the method of [25] can not be used in. In the third section,
by using this method, we then further consider the corresponding nonlinear Cauchy
problem in both of the Maxwellian and nonMaxwellian molecules cases. Under some
suitable assumptions, we also give a positive answer successfully for the Gevrey regularity
of the solutions.
Finally, Chapter 4 is a summary of the thesis, mainly summarize the results , the
deficiencies existing and the further research direction.
Keywords: Cauchy integral theorem; DebyeYukawa potential; Gevrey class regularity;
Inverse power law potential; NonMaxwellian molecules; Noncutoff; Pseudodifferential
operators.
 Differential equations, Partial