双曲平衡律系统半整体熵解的适定性及其应用

项目名称： 双曲平衡律系统半整体熵解的适定性及其应用

项目编号： No.11501122

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 余磊

作者单位： 复旦大学

项目金额： 18万元

中文摘要： 申请人计划研究具有一般形式的一维双曲平衡律系统在有限区域上的初边值问题的半整体熵解（有界变差足够小，满足熵条件的弱解）的适定性：任意给定一固定的有限时间区间，是否只要初边值的有界变差相应地足够小（依赖于时间段大小），在这一时间区间内熵解在所考虑的函数空间内有存在唯一性和稳定性。一般在没有耗散边界条件和外源项函数对角占优的假设下，不存在整体熵解，从而半整体熵解将可能是最优的结果。很多实际问题只用考虑有限时间内解的情况，所以半整体熵解的适定性理论有其广泛的应用价值。. 为了更好地驱动和完善半整体熵解理论，申请人计划应用半整体熵解的适定性理论研究双曲平衡律系统熵解的边界能控性。寻找能实现此系统边界能控所要满足的合理条件，以及具有广泛适用性的证明方法。结合具体的双曲平衡律模型，研究其熵解的单侧边界能控性，以填补这方面科研成果的空白。

中文关键词： 双曲守恒律组；平衡律；半整体熵解；适定性；边界能控性

英文摘要： I plan to study well-posedness of semi-global entropy solutions (weak solutions with small total variation, satisfying entropy condition) to general hyperbolic systems of balance laws, that is, given any fixed finite time interval, if the total variation of initial and boundary data is small enough, does the entropy solution uniquely exist in the admissible function space on the time interval with stability? In general, without the dissipative boundary condition and diagonal dominated assumption on the source term, there is no global entropy solution, therefore, the existence of semi-global entropy solution would be the optimal result. Since in many real applications, solutions only need to be considered on the finite time interval, the theory of semi-global entropy solutions has its value of applications.. In order to motivate and improve the study of semi-global entropy solution, I plan further to apply the well-posedness of semi-global entropy solutions to study boundary controllability of hyperbolic balance laws. I will try to find under what condition the system is controllable through boundary control and the general method to solve boundary controllability problem in the context of entropy solutions. Also I plan to solve the one side boundary controllability of entropy solution, by considering the model of hyperbolic balance laws, which is poorly known up to now.

英文关键词： hyperbolic conservation laws;balance laws;semi-global entropy solution;well-posedness;boundary controllability