一阶拟线性双曲型方程组慢时间尺度下的零松弛极限  

Zero relaxation limit in slow time scaling for first-order quasi-linear hyperbolic systems

作　　者：彭跃军 PENG YueJun

机构地区：[1]Laboratoire de Mathématiques Blaise Pascal,Université Clermont Auvergne,CNRS

出　　处：《中国科学：数学》2017年第10期1255-1276,共22页Scientia Sinica：Mathematica

摘　　要：本文考虑慢时间尺度下带松弛时间源项的高维一阶拟线性双曲型方程组Cauchy问题的光滑解,这个方程组具有非守恒的形式;假设它是部分耗散的对称双曲组,当松弛时间趋于零时,它在形式上趋于一个两阶非线性抛物型方程组.在双曲型方程组满足一些结构性的假设下,本文得到了两个收敛性的结果.对于大初值,本文证明了双曲型方程组在一个对松弛时间一致的时间区间上的收敛性,当初值在常数平衡态附近变化时,证明了光滑解关于时间的一致整体存在性和双曲型方程组的整体收敛性.本文也给出一些有物理背景的例子来作为这些结果的应用.In this paper,we consider smooth solutions to the Cauchy problem for a multidimensional firstorder quasi-linear hyperbolic system with a small parameter called relaxation time.The system is written in non-conservative form in a slow time scaling.We suppose that it is symmetrizable hyperbolic and partially dissipative.As the parameter goes to zero,it converges formally to a second-order nonlinear parabolic-type system.Under additional structural conditions on the hyperbolic system,we justify this convergence at two levels.For large initial data,we prove the local-in-time convergence of the system in a uniform time interval with respect to the parameter.When the initial data stay in a neighborhood of an equilibrium state,we prove the uniform global existence of solutions and the global-in-time convergence of the system.We also give examples of physical models to which the above results can be applied.

关 键 词：一阶拟线性双曲组 零松弛极限 抛物型方程组 局部和整体收敛性 

分 类 号：O175.27[理学—数学]