论文标题
在关键的规律性设置中部分耗散系统和强大的放松极限
Partially dissipative systems in the critical regularity setting, and strong relaxation limit
论文作者
论文摘要
许多物理现象可以通过一阶双曲方程进行建模，其耗散性或扩散术语。例如，在气体动力学中，这种情况是在进化过程中保守质量的情况，但是动量平衡包括扩散（粘度）或阻尼（松弛）项，或者在数值模拟中，通过松弛方案对保护定律进行了模拟。 S.K.首先指出了这种所谓的部分耗散系统。戈杜诺夫（Godunov）在1961年在俄罗斯人的简短说明中。在1984年，S。Kawashima在他的博士学位论文中强调了一个简单的标准，确保了在线性稳定状态的附近存在全球强大解决方案。该标准已在许多研究工作中重新审视。特别是，K。Beauchard和E. Zuazua在2010年提出了一种构建Lyapunov功能的明确方法，允许完善Kawashima的结果并确定全球存在的结果，从而在某些情况下未涵盖的某些情况。这些笔记基本上源自T. crin-barat的博士学位论文，该论文最初是由Y. Giga和A. Novotn {}}授予的手册的一章中早期观察到作者的动机。我们的主要目的是将Beauchard和Zuazua的方法调整为一类可对称的准线性双曲系统（包含可压缩的Euler方程），并在关键的规则性设置中，可以跟踪相对于例如。放松参数。与Beauchard和Zuazua的作品相比，我们展示了一种“阻尼模式”，该模式将在具有关键规律性的全球解决方案的构建中起关键作用，以证明最佳的时间预期估计值，最后但并非最不重要的一点是，在研究强烈的放松限制方面。为了简单起见，我们在这里专注于一类简单的部分耗散系统，但是总体策略相当灵活，并且可以适应更多涉及的情况。
Many physical phenomena may be modelled by first order hyperbolic equations with degenerate dissipative or diffusive terms. This is the case for example in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term, or, in numerical simulations, of conservation laws by relaxation schemes. Such so-called partially dissipative systems have been first pointed out by S.K. Godunov in a short note in Russian in 1961. Much later, in 1984, S. Kawashima highlighted in his PhD thesis a simple criterion ensuring the existence of global strong solutions in the vicinity of a linearly stable constant state. This criterion has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima's results and to establish global existence results in some situations that were not covered before. These notes originate essentially from the PhD thesis of T. Crin-Barat that was initially motivated by an earlier observation of the author in a Chapter of the handbook coedited by Y. Giga and A. Novotn{ý}. Our main aim is to adapt the method of Beauchard and Zuazua to a class of symmetrizable quasilinear hyperbolic systems (containing the compressible Euler equations), in a critical regularity setting that allows to keep track of the dependence with respect to e.g. the relaxation parameter. Compared to Beauchard and Zuazua's work, we exhibit a 'damped mode' that will have a key role in the construction of global solutions with critical regularity, in the proof of optimal time-decay estimates and, last but not least, in the study of the strong relaxation limit. For simplicity, we here focus on a simple class of partially dissipative systems, but the overall strategy is rather flexible, and adaptable to much more involved situations.
