Compressible Euler equations with transport noise

具有传输噪声的可压缩欧拉方程

• 批准号: 525608987
• 负责人: Professor Dr. Dominic Breit
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/525608987?language=en
• 关键词: Compressible; Euler; equations; transport; noise

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.

我们考虑了具有输运类型随机扰动的多维等熵可压缩欧拉方程。双曲线传输和随机性的相互作用是SPP-2410的一个中心问题。一方面，传输噪声是由湍流理论中的物理模型驱动的。另一方面，最近的研究表明，这种类型的噪声可以产生正规化效应。我们的目标是一个相当完整的关于随机偏微分方程组的适定性(和病态)的图景。作为第一步，我们计划证明耗散测值鞅解的存在性(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、它们的长时间行为以及马尔可夫选择的存在性。第二个工作包涉及弱解的存在(其中非线性用函数而不是度量来描述)。基于凸积分的方法，我们希望证明对任意初始数据都存在无穷多个解析弱解。这些解在概率意义上将是强的，即它们存在于给定的随机基础上。然而，它们并不满足能源不平等的要求。最后，我们将分析运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。