二维玻尔兹曼方程的不可压缩极限
Incompressible Limit of the Two Dimensional Boltzmann Equation
摘要: 本文拟研究二维空间区域上玻尔兹曼方程的不可压缩Navier-Stokes-Fourier极限。由于有界区域上的玻尔兹曼方程的解没有高阶正则性，故本文拟采用最新的L2-L∞方法并结合解的宏观部分的L4估计，来获取余项方程线性部分的一致上界估计，进而通过迭代方法得到余项方程解的存在性，最后得出原玻尔兹曼方程解的存在性和收敛极限。
Abstract: In this paper, we study incompressible Navier-Stokes-Fourier limit of the two dimensional Boltzmann equation. The solution of the Boltzmann equation has no high order regularity in the bounded region, so we use a recent quantitative L2-L∞ approach with a new L4 estimate for the hydrodynamic part, to obtain uniform upper estimation of the linear part of remainder equation, and then obtain the existence of the solution of remainder equation through iteration. Finally, we get existence of the solution of the Boltzmann equation and the convergence limit.
文章引用：高婷婷. 二维玻尔兹曼方程的不可压缩极限[J]. 理论数学, 2020, 10(2): 128-138. https://doi.org/10.12677/PM.2020.102019

参考文献

[1] Guo, Y. (2010) Decay and Continuity of the Boltzmann Equation in Bounded Domains. Archive for Rational Mechanics and Analysis, 197, 713-809. [Google Scholar] [CrossRef] 
[2] Bardos, C., Golse, F. and Levermore, D. (1991) Fluid Dynamic Limits of Kinetic Equations I: Formal Derivations. Journal of Statistical Physics, 63, 323-344. [Google Scholar] [CrossRef] 
[3] Bardos, C., Golse, F. and Levermore, D. (1993) Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Communications on Pure and Applied Mathematics, 46, 667-753. [Google Scholar] [CrossRef] 
[4] Golse, F. and Saint-Raymond, L. (2004) The Navier-Stokes Limit of the Boltzmann Equation for Bounded Collision Kernels. Inventiones Mathematicae, 155, 81-161. [Google Scholar] [CrossRef] 
[5] Golse, F. (2005) Hydrodynamic Limits. European Congress of Mathematics. Eur. Math. Soc, 699-717. [Google Scholar] [CrossRef] 
[6] Esposito, R., Guo, Y., Kim, C. and Marra, R. (2018) Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit. Annals of PDE, 4, 1-55. [Google Scholar] [CrossRef] 
[7] Leoni, G. (2009) A First Course in Sobolev Spaces. AMS Graduate Studies in Mathematics, 105. [Google Scholar] [CrossRef] 
[8] Krylov, N.V. (2008) Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. AMS Graduate Studies in Mathematics, 96. [Google Scholar] [CrossRef] 

为你推荐