Abstract of Proposed Research Michael Westdickenberg
This project is to investigate certain properties of the solutions of the isentropic Euler equations. They are a system of hyperbolic conservation laws that model the dynamics of compressible fluids. Part 1 of the project will investigate the global existence of spherically symmetric solutions of the multi-dimensional isentropic Euler equations, assuming only the natural bounds of finite mass and total energy. A compactness framework will be developed that can also be applied to other situations where the geometry of the problem plays an important role, such as nozzle flows and shallow water flows over complex topographies. It is well-known that the main difficulty in the analysis of hyperbolic conservation laws comes from the occurrence of jump discontinuities, ie shocks. The second and third part of this project will therefore focus on issues related to shock formation. After a shock appears, the important issue is how does the solution continue? Since weak solutions are typically non-unique, a selection criterion is required to single out the relevant behavior among all possible continuations. Part 2 of this project will propose a new variational selection principle for the multi-dimensional isentropic Euler equations and study its implications. The selection principle is inspired by the interpretation of certain parabolic equations as abstract gradient flows on spaces of probability measures. Part 3 of this project will probe the regularity of weak solutions of conservation laws, both in terms of the function spaces to which they belong, and in terms of the fine structure of singularities. One goal is to show that shocks are located in a codimension-one rectifiable set, and that the conserved quantities admit strong traces along the shock set.
Important processes in both the natural sciences and engineering can be modeled by multi-dimensional systems of hyperbolic conservation laws. They are first order partial differential equations in divergence form. Typically these processes exhibit some sort of propagation and interaction of waves and conservation of certain quantities such as mass or momentum. Since these phenomena are ubiquitous in nature, systems of hyperbolic conservation laws can be found in fields as diverse as physics, geophysics, environmental sciences, meteorology, epidemiology, and biology. In this project, we will study the particular equations which model the dynamics of inviscid, compressible fluids, and which are a system of hyperbolic conservation laws. These equations have a long and rich history starting with the work of Daniel Bernoulli and Leonard Euler in the eighteenth century. Despite their enormous importance for both theory and applications, and the fact that their numerical treatment has reached a high level of sophistication, the mathematical theory of the Euler equations is still incomplete. It is the goal of this project to better understand this fundamental system of conservation laws.
