The Isentropic Euler Equations and Optimal Transport

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The project will explore the deep connections between compressible fluid dynamics and the theory of optimal transport. It will establish that the isentropic Euler equations can be understood as a steepest descent on a suitable space of probability measures, by proving that a sequence of approximate solutions generated by a certain variational time discretization converges to a measure-valued solution of the conservation law. For the particular cases of one space dimension and of self-similar solutions, it will be shown that these measure-valued solutions are weak entropy solutions in the classical sense. A novel solution concept for the isentropic Euler equations will be explored in the framework of second-order differential inclusions. The interpretation as a differential inclusion will also be applied to one-dimensional models of compressible fluid flows with interactions, in the context of sticky particle dynamics. Stability of sticky particle solutions will be shown and global existence will be established. The variational time discretization will be used to derive new numerical methods for the two-dimensional isentropic Euler equations. Finally, the project will investigate the fine structure of solutions through gamma-convergence of the energy dissipation functional.The compressible Euler equations model the dynamics of compressible fluids, in particular, of gases. They express in mathematical terms a simple but fundamental principle of physics: that mass and energy must be preserved. They form the basic building block for mathematical models in various disciplines of science and engineering. One example is aerodynamics: an understanding of the properties of solutions to the compressible Euler equations are necessary to design more efficient airplane wings or cars with smaller aerodynamical resistance. Other examples include the theory of combustion processes, weather and climate modeling, models of blood flows, and more. This project will advance the mathematical understanding of the compressible Euler equations by harnessing tools from another field in mathematics that has attracted a lot of interest recently and is developing rapidly: optimal transport. This theory grew out of a very simple engineering problem: How does one move an amount of sand or debris from one place to another, with minimal effort? When applied to the compressible Euler equations, this theory expresses the idea that typically nature tries to do things in the most economical way. This point of view allows us to gain new insights into the nature of the compressible Euler equations and the properties of their solutions.