The Isentropic Euler Equations and Optimal Transport

该项目将探索可压缩流体动力学和最佳传输理论之间的深层联系。它将建立等熵欧拉方程可以被理解为一个最陡的下降到一个合适的概率测度空间，通过证明，由一定的变分时间离散化产生的近似解序列收敛到守恒律的测度值解。对于一维空间和自相似解的特殊情况，将证明这些测度值解是经典意义下的弱熵解。一个新的解决方案的概念，等熵欧拉方程将探讨的框架内的二阶微分包含。作为一个微分包含的解释也将适用于一维模型的可压缩流体流动的相互作用，在粘性粒子动力学的背景下。粘性粒子解的稳定性将被证明，并建立整体存在性。变分时间离散方法将被用于推导二维等熵欧拉方程的新的数值方法。最后，该项目将通过能量耗散泛函的伽玛收敛研究解的精细结构。可压缩欧拉方程模拟可压缩流体，特别是气体的动力学。它们用数学术语表达了一个简单但基本的物理学原理：质量和能量必须守恒。它们构成了科学和工程各个学科中数学模型的基本构建块。一个例子是空气动力学：对可压缩欧拉方程解的性质的理解对于设计具有更小空气动力学阻力的更有效的飞机机翼或汽车是必要的。其他例子包括燃烧过程理论，天气和气候建模，血液流动模型等。该项目将通过利用数学中另一个领域的工具来推进对可压缩欧拉方程的数学理解，该领域最近引起了很多兴趣并且正在迅速发展：最佳运输。这个理论源于一个非常简单的工程问题：如何用最小的努力将大量的沙子或碎片从一个地方移到另一个地方？当应用于可压缩欧拉方程时，这个理论表达了这样一种思想，即自然界通常试图以最经济的方式做事。这一观点使我们对可压缩欧拉方程的性质及其解的性质有了新的认识。

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.