带外场的Boltzmann方程Cauchy问题的强解-应用数学专业论文.docxVIP

华中科技大学硕士学位论文 华 中 科 技 大 学 硕 士 学 位 论 文 II II Abstract In the Kinetic Theory, The Boltzmann equations is the most basic and classic model. The research for the Boltzmann equations has been one of the most important and challenging field in Partial Differential Equations because of its rich physical background and practical applications. Thus, it is very important to reveal the properties of the Boltzmann equations mathematically. In this paper, we study the existence and propagation of regularity of strong solutions to Cauchy problem for Boltzmann equation with an external field for initial 0data f (x, v) ?W1, p 0 (R3 ??R3 ) with 1 ??p ????, and the divergence of the external field being zero, i.e.  ?v ??F(t, x, v) ??0 , and  B( u ,??) ??u b ?u? ???????C(1??u )  ? ?, whe ? ? ?2 ??????0 . In order to solve this problem,we prove our main conclusions by asymptotic methods. Firstly, we modify the initial data to obstain the approximating solutions, then get the classical solutions by taking advantage of Guo’ results. Secondly, we obstain some uniform estimates of classical solutions, then get Lp strong compactness of the classical solutions by taking advantage of these uniform estimates and Kolmogorov-Riesz theorem. Finally, we get the limit solutions by taking advantage of compactness and get propagation of regularity of strong solutions. Compared with Guo’s results,we weaken initial data from f (x, v) ?C1 (R3 ??R3 ) to f (x, v) ?W1, p (R3 ??R3 ) 00?1 ??p ?????and get propagation of regularity of strong solutions. Thus, in this paper, the discussion for the existence and propagation of regularity of strong solutions to Cauchy problem for Boltzmann equation with an external field is a good supplement of Guo’s results. 0 0 The research for the Cauchy problem the Boltzmann equations with an external field is of great theoretical and practical significance. But, so far, there are also many open PAGE IV PAGE IV questions to be researched and explored,