Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws
Large Time Behavior of Solutions for General Quasilinear Hyperbolic-Parabolic Systems of Conservation Laws
复制标题
DOI：
10.1090/memo/0599
复制
发表时间：
1997-07
期刊：
影响因子：
--
通讯作者：
Tai-Ping Liu;Yanni Zeng
中科院分区：
文献类型：
--
作者：
Tai-Ping Liu;Yanni Zeng
我们对粘性守恒律的解的时间渐近行为很感兴趣。通过线性化系统格林函数的逐点估计和非线性扩散波的耦合分析，得到了解的时间渐近性态的显式表达式。这产生了积分范数中的最优估计。对于大多数物理模型，粘性矩阵不是正定的，系统是双曲线-抛物线，而不是一致抛物线。这意味着格林函数可能包含狄拉克[小写希腊]Delta函数。当相应的无粘系统为非严格双曲系统时，时间渐近态包含广义Burgers解。将我们的一般理论应用于可压缩的Navier-Stokes方程和磁流体动力学方程，对这些问题进行了说明。
We are interested in the time-asymptotic behavior of solutions to viscous conservation laws. Through the pointwise estimates for the Green's function of the linearized system and the analysis of coupling of nonlinear diffusion waves, we obtain explicit expressions of the time-asymptotic behavior of the solutions. This yields optimal estimates in the integral norms. For most physical models, the viscosity matrix is not positive definite and the system is hyperbolic-parabolic, and not uniformly parabolic. This implies that the Green's function may contain Dirac [lowercase Greek] Delta-functions. When the corresponding inviscid system is non-strictly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. These are illustrated by applying our general theory to the compressible Navier-Stokes equations and the equations of magnetohydrodynamics.