带非局部耗散项的守恒律方程大扰动解的整体存在性  

Global existence of large solutions to conservation laws with nonlocal dissipation-type terms

作　　者：王利娟 王维克 徐鑫 Lijuan Wang;Weike Wang;Xin Xu

机构地区：[1]上海对外经贸大学统计与信息学院,上海201620 [2]上海交通大学数学科学学院,上海200240 [3]北京应用物理与计算数学研究所,北京100084

出　　处：《中国科学：数学》2018年第5期589-608,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11771248);中国博士后科学基金(批准号:2017M610818)资助项目

摘　　要：本文考虑带非局部耗散项的单个守恒律方程大扰动解的整体存在性.首先,针对方程次临界和临界两种不同的情形,利用Green函数方法和环形分解的技巧,构造开放式高频估计方法,得到了一个新的解的正则性准则.然后,利用极大值原理得到方程解的极大模的有界性,验证了次临界情形下解满足相应的正则性准则.对于临界这一更困难的情形,本文应用非线性极大值原理方法得到了更好一点的有界性估计,验证了临界情形下解也满足相应的正则性准则,从而得到了Cauchy问题大扰动经典解的整体存在性.We consider the global existence for large perturbation solutions of scalar conservation laws with nonlocal dissipative structures. At the very beginning, based on the Green function method and the LittlewoodPaley decomposition, we establish a new regularity criterion to improve the regularity of the solution. It is worth to point out that the key points in the regularity criterion are that ∥u∥L∞ ＋∞ for 0≤ s 1 and ∥u∥C~α ＋∞ for s = 1. Thus, in the following part we devote ourselves to verify the two key points. For the subcritical case, we obtain the boundedness of solution in Lpfor any p ∈ [2, ＋∞） by the maximum principle. Thanks to the nonlinear maximum principle, C^α boundedness is established for the critical case. Finally, the global existence of classical solutions to the scalar conservation with large initial data is obtained.

关 键 词：单个守恒律方程 非局部耗散项 Cauchy问题 极大值原理 非线性极大值原理 

分 类 号：O175[理学—数学]