Some Research About The Regularity Of The Solutions For Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

As one of the most interesting and important problems for physicists and math-ematicians, the regularity research for the solution of the Boltzmann equation has at-tracted many people in recent years, its physical significance is to show the smoothness property of the one-particle distribution function for the simple monatomic gas. In the early study of this area, to avoid the mathematical difficulty from the grazing effects in elastic collisions between particles, most of the works on the Boltzmann equation are based on Grad's cutoff assumption, the study which under the assumption shows that in weighted Lp space, the solution preserves the same regularity as the initial data. On the other hand, it is well know that when Grad's angular cutoff is not assumed, the Boltz-mann 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 class regularity of the solution of the Boltzmann equation without angular cutoff, so far there have many advanced research results on this topic hence 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 non-Maxwellian molecules case with the Debye-Yukawa potential, under the condition of Lipschitz con-tinuity in the velocity variable, the positive weak solution of the spatially homogeneous Boltzmann equation lies in the Sobolev space Hloc+∞(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 con-sider the corresponding linearized Cauchy problem in the non-Maxwellian 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, in-cluding not only the abstract pseudo-differential 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 non-Maxwellian 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 non-Maxwellian molecules cases. Un-der 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.