Periodic and Large Amplitude Solutions for the compressible Euler equations

该项目的重点是非线性波在无粘欧拉方程中的传播，该方程描述了连续介质中的质量，动量和能量守恒。 解决方案的主要特点是存在的冲击波，这提出了物理和数学上的困难。 两个基本问题被认为是：第一个是扩展的Glimm-Lax的存在和衰减理论的解决方案具有大幅度。 这个问题需要多波相互作用的分析，包括真空，系统是奇异的。 第二个问题是关于不耗散能量的无激波周期解的存在性。 这是由非线性波的多次反射和非线性叠加原理引起的，并导致小因子问题。激波的特征是介质特性的突然的、几乎不连续的变化。在激波运动中，例如在空气中，压力、温度和流量总是非常迅速地上升。冲击波在许多物理系统中自然发生，并且与溶液的衰变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振幅解的这种衰减，忽略了高阶效应，如粘性和热损失;该理论通常被认为在更广泛的背景下成立。 本项目的第一部分扩展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令人惊讶的结论，即存在不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，在较大的系统中，多重波反射效应减缓了冲击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.