浦赟,张永前.欧拉 - 玻尔兹曼方程整体光滑解[J].数学年刊A辑,2022,43(2):119~136
欧拉 - 玻尔兹曼方程整体光滑解
Global Existence of Smooth Solutions to the Euler-Boltzmann Equations
投稿时间:2021-07-27  修订日期:2022-01-14
DOI:10.16205/j.cnki.cama.2022.0009
中文关键词:  欧拉 - 玻尔兹曼方程, 柯西问题, 局部适定性, 能量估计, 整体光滑解
英文关键词:Euler-Boltzmann equations, Cauchy problem, Local wellposedness,Energy estimates, Global smooth solutions
基金项目:
作者单位
浦赟 复旦大学数学科学学院, 上海 200433 
张永前 复旦大学数学科学学院, 上海 200433 
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中文摘要:
      本文证明了Rd 中具有某一类小初值的等熵欧拉 - 玻尔兹曼方程整体光滑解的存在性.本文首先构造了等熵欧拉 - 玻尔兹曼方程的局部解, 并证明了局部解的适定性. 此外,文中还构造了关于原方程的随时间 t 增加、具有良好的衰减性质的整体光滑背景解. 同时, 当方程的辐射项系数满足一定条件时, 本文建立了关于源项的估计.通过将背景解的衰减与源项的估计结合起来, 文中证明了存在整数 s>d/2 + 1 ,使得背景解与原方程解的 Hs(Rd)x L2(R+ x Sd-1;Hs(Rd))范数之差始终是有界的, 从而保证了原方程整体光滑解的存在性.
英文摘要:
      The authors are concerned with the global existence of smooth solutions to the isentropic Euler-Boltzmann equations in Rd, with a class of small initial data. Firstly, they construct local solutions to the problem and then the wellposedness of the local solutions is proved. Also, they construct global smooth background solutions to the approximate system, which possess a nice decay as t increases. Meanwhile, estimates of the source terms are established with some assumptions on the coefficients of the radiation part. Combining the decay of the background solutions and the estimates on the source terms, the authors show that for some integer s > d/2 + 1, the difference of Hs (Rd) × L2(R+ × Sd-1; Hs(Rd))norm between approximate solutions and real solutions remains finite, which ensures the global existence of smooth solutions to original problems.
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