Compressible Euler equations with transport noise

本文研究了具有输运型随机扰动的多维等熵可压缩Euler方程。双曲线传输和随机性的相互作用是SPP-2410的核心问题。一方面，传输噪声是由湍流理论中的物理模型激发的。另一方面，最近已经表明，这种类型的噪声可以具有正则化效应。我们的目标是在一个相当完整的图片有关的适定性（和不适定性）的基本系统的随机偏微分方程。作为第一步，我们计划证明耗散测度值鞅解的存在性（其中概率空间不是先验给定的，而是问题的一部分）。最后，我们将分析它们的一些性质，如弱-强唯一性，它们的长时间行为和马尔可夫选择的存在性。第二个工作包关注弱解的存在性（其中非线性由函数而不是测度描述）。利用凸积分的方法，我们希望证明在任意初始条件下存在无穷多个解析弱解。这些解在概率意义上是强的，即，它们存在于给定的随机基础上。但是，它们不满足能量不等式。最后，我们将分析传输噪声是否可以延迟可压缩欧拉方程解的不可避免的爆破。

We consider the isentropic compressible Euler equations in multi-dimensions with stochastic perturbation of transport type. The interplay of hyperbolic transport and randomness is a central question in the SPP-2410. On the one hand, transport noise is motivated by the physical modelling in turbulence theory. On the other hand, it has been shown recently that this type of noise can have regularising effects. We aim at a rather complete picture concerning the well-posedness (and ill-posedness) of the underlying system of stochastic PDEs. As a first step, we plan to prove existence of dissipative measure-valued martingale solutions (where the probability space is not a priori given but part of the problem). Eventually, we will analyse some of their properties such as weak-strong uniqueness, their long-time behaviour and the existence of Markov selections. A second working package is concerned with the existence of weak solutions (where the nonlinearities are described by functions rather than measures). Based on the method of convex integration we hope to prove the existence of infinitely many analytically weak solutions for any initial datum. These solutions will be strong in the probabilistic sense, i.e., they exist on a given stochastic basis. However, they do not satisfy an energy inequality. Finally, we will analyse whether a transport noise can delay the otherwise inevitable blow-up of solutions to the compressible Euler equations.