Strong Solutions To The Cauchy Problem For Boltzmann Equations With An External Field

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 data fo(x,v)∈W1,p(R3×R3) with1<p<∞, and the divergence of the external field being zero, i.e. Vv▽v·F(t,x,v)=0, and B(|u|,θ)=|u|λb(u·σ)≤C(1+|u|λ), where-2<λ<≤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 fo(x,v)∈C1(R3×R3) to fo(x,v)∈W1,p(R3×R3)(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.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 questions to be researched and explored, such as the situation of p1in this article orthe existence and properties of classical solutions to Cauchy problem for Boltzmannequation with an external field. Thus, we expect these problems could be improved andbreakthroughed in the theoretical research.