Periodic and Large Amplitude Solutions for the compressible Euler equations

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

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.