Dissipative weak solutions for non-viscous single and multicomponent/phase flows

气体动力学的欧拉方程用于许多科学和工程问题-包括空气流动的描述-并涉及质量，动量和总能量守恒。长期以来，人们并不清楚多维欧拉方程在弱熵解类中是否适定，即是否存在唯一的解，该解连续依赖于初始数据。这已经在一些研究论文中被证明是错误的，并且耗散弱（DW）解被引入作为经典解概念的推广。它们已经成为一种有意义的解决方案概念。 一些数值方法已经研究了DW解的收敛性。重要的是，数值方法确保欧拉方程的结构特性，例如，密度和压力的正性。这是项目的起点。首先，进行统一的收敛性分析。特别是，结构保持的方法进行检查，并证明了DW解决方案的收敛性。对于各种有限元方法，仅在半离散情况下证明了收敛性，即时间保持连续。在项目内部，我们进一步分析了完全离散情况下的收敛性，以缩小这一差距。在项目的第二部分，我们专注于非粘性多组分/相流。它们用于描述复杂的模型，包括欧拉方程与附加耦合项的组合。 对于欧拉方程的收敛性研究，数值方法的结构保持特性已经是必不可少的。在这一部分中，新的结构保持计划的非粘性多组分/相流将开发和分析更详细。在该项目的最后一部分，欧拉方程的DW解决方案的概念将扩展到多组分/相模型，理论上检查和一致的解决方案理论将提供。随后，（新的）结构保持方法的DW解决方案的收敛性进行了分析。该项目的目的是对欧拉方程的DW解进行一致收敛性分析，构造新颖的结构保留方法，并在更一般的模型中扩展和建立DW的概念。

The Euler equations of gas dynamics are used for many scientific and engineering problems - including the description of air flows - and involve the conservation of mass, momentum and total energy. For a long time, it was unclear whether multidimensional Euler equations are well-posed in the class of weak entropy solutions, i.e. whether there exists a unique solution that is continuously dependent on the initial data. This has now been disproven in several research papers and dissipative weak (DW) solutions were introduced as a generalization of the classic solution concept. They have established themselves as a meaningful solution concept. Some numerical methods have already been investigated with regard to their convergence properties for DW solutions. It was important that the numerical methods ensure the structural properties of Euler equations, e.g. positivity of density and pressure. This is the starting point of the project. First, a unified convergence analysis is performed. Especially, structure-preserving methods are examined and convergence properties with regard to DW solutions are demonstrated. For various finite element methods, convergence has only been proven in the semi-discrete case, i.e. the time was kept continuous. Inside the project, we further analyze convergence in the fully discrete case to close this open gap. In the second part of the project, we focus on non-viscous multicomponent/phase flows. They are used to describe complex models and include combinations of Euler equations with additional coupling terms. Already for convergence investigations regarding the Euler equations, the structure-preserving properties of the numerical procedure have been essential. In this part, novel structure-preserving schemes for non-viscous multicomponent/phase flows will be developed and analyzed in more detail. In the final part of the project, the concept of the DW solutions of the Euler equations will be extended to multicomponent/phase models, theoretically examined and a consistent solution theory will be provided. Subsequently, the convergence properties of the (novel) structure-preserving methods with regard to DW solutions are analyzed. The aim of the project is to give a uniform convergence analysis regarding DW solutions for the Euler equation, to construct novel structure-preserving methods and extend and establish the concept of DW in more general models.