Viscous shocks and long-time behavior of scalar conservation laws
Viscous shocks and long-time behavior of scalar conservation laws
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DOI：
10.3934/cpaa.2023119
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发表时间：
2023-06
影响因子：
1
通讯作者：
T. Gallay;A. Scheel
中科院分区：
文献类型：
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作者：
T. Gallay;A. Scheel
我们通过$\omega$-极限集的结构研究了标量粘性守恒律的长时间行为。我们发现，$\omega$-极限集总是包含常数或冲击建立收敛到冲击任意单调的初始数据。在特定的情况下，伯格斯方程，我们审查和完善的结果，参数化整个解决方案的概率措施，我们构建的初始数据的$\omega$-极限集不减少到一个单一的冲击的翻译。最后，我们提出了几个开放的问题有关的描述长时间的动力学。
We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $\omega$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.