Periodic and Large Amplitude Solutions for the compressible Euler equations

本项目主要研究无粘欧拉方程中的非线性波的传播，该方程描述了连续介质中的质量、动量和能量守恒。解的主要特征是存在冲击波，这既有物理上的困难，也有数学上的困难。考虑了两个基本问题：一是Glimm-lax存在性和衰减性理论推广到大振幅解。这个问题需要对包括真空在内的多波相互作用进行分析，在真空中，系统是奇异的。第二个问题涉及不耗散能量的无激波周期解的存在性。这些波动是由非线性波的多次反射和非线性叠加原理引起的，并导致小因子的问题。冲击波的特征是介质特性的突然、几乎不连续的变化。在激波运动中，例如在空气中，压力、温度和气流总是会极其迅速地上升。冲击波自然地出现在许多物理系统中，并与溶液的衰减和能量的耗散密切相关；衰减冲击波的一个众所周知的表现是音爆。著名的Glimm-Lax理论精确地描述了两个方程的小幅度解的这种衰减，忽略了粘性和热损失等更高阶的影响；该理论通常被认为在更广泛的背景下适用。本项目的第一部分推广并证实了由两个方程组描述的任意强度波的Glimm-Lax理论。该项目的第二部分揭示了一个令人惊讶的结论，即存在不会形成冲击波的解决方案，因此不会衰减。特别是，这项研究表明，在更大的系统中，多次波反射效应减缓了激波的形成和随后的能量耗散。这些效应的可控性将产生许多后果，包括应用于远程信号和飞机设计。

This project focuses on the propagation of nonlinear waves in the inviscid Euler equations, which describe conservation of mass, momentum and energy in a continuous medium. The dominant feature of solutions is the presence of shock waves, which present both physical and mathematical difficulties. Two fundamental problems are considered: first is the extension of the Glimm-Lax existence and decay theory to solutions having large amplitude. This problem necessitates an analysis of multiple wave interactions up to and including the vacuum, where the system is singular. The second problem concerns the existence of shock-free periodic solutions which do not dissipate energy. These arise from multiple reflections of nonlinear waves together with a nonlinear superposition principle, and lead to problems of small divisors.Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium. Across a shock moving, for example in the air there is always an extremely rapid rise in pressure, temperature and of the flow. Shock waves occur naturally in many physical systems, and are closely associated with the decay of solutions and dissipation of energy; a well-known manifestation of a decaying shock wave is a sonic boom. The celebrated Glimm-Lax theory precisely describes this decay for small amplitude solutions of two equations, neglecting higher order effects such as viscosity and heat loss; the theory is routinely assumed to hold in much wider contexts. The first part of this project extends and confirms the Glimm-Lax theory for waves of arbitrary strength described by systems of two equations. The second part of the project reveals the surprising conclusion that there are solutions which do not form shock waves and thus do not decay. In particular, this research indicates that in larger systems multiple wave reflection effects slow down the formation of shocks and the subsequent dissipation of energy. Controllability of these effects would have many consequences, including applications to long-range signaling and airplane design.