非凸粘弹性系统中由非经典Liu激波与稀疏波组成的复合波的稳定性
Stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for a non-convex viscoelastic system
AI总结：
本文研究非凸粘弹性系统中由非经典Liu激波与稀疏波组成的复合波的时间渐近稳定性，采用a-收缩方法完成证明，为该类复合波的稳定性提供了首个相关结果。
摘要：
我们研究非凸粘弹性系统中由非经典Liu激波与稀疏波组成的复合波的时间渐近稳定性。通过为粘性激波选取精细的随时间变化的平移函数，并为相对熵选取合适的权重函数，我们成功采用a-收缩方法证明了：对于具有三次非凸应力-应变关系的一维粘弹性系统，来自两个不同特征族的粘性Liu激波与稀疏波的复合波具有时间渐近稳定性。此处的非经典Liu激波可为退化或非退化情形，可视为标量Oleinik激波的系统版本。需注意，近退化与完全非退化这两种情形下，平移函数与权重函数的选取存在显著差异。本文首次给出了非凸物理系统中由来自两个不同族的非经典Liu激波与稀疏波组成的复合波的时间渐近稳定性结果。
英文摘要：
We investigate the time-asymptotic stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for non-convex viscoelastic system. By choosing delicate time-dependent shift functions to the viscous shock and suitable weight functions for the relative entropy, we succeed in proving the time-asymptotic stability of the composite of a viscous Liu shock and a rarefaction wave from two different characteristic families for the one-dimensional viscoelastic system with a cubic non-convex stress-strain relation by $a$-contraction method. Here the non-classical Liu shock can be either degenerate or non-degenerate, which can be seen as the system version of the scalar Oleinik shock. Note that both the shift and the weight functions are chosen quite differently in the two cases of nearly degenerate and fully non-degenerate. The paper represents the first time-asymptotic stability result for the composite waves consisting of both non-classical Liu shock and rarefaction wave from two different families for the non-convex physical system.
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