Koordinationsantrag

非线性双曲平衡定律在流体力学过程的模拟中是普遍存在的。它们能够开发强大的数值模拟方法，为关键应用提供决策支持，例如硅气和航天器设计或气候变化研究。然而，关于独特的双曲线特征的基本问题仍然悬而未决，包括激波和横波的多尺度干扰，或者双曲线传输和随机环境的相互作用。多维无粘流方程在很大程度上没有解决的适定性问题与高雷诺数下的湍流运动规律密切相关。进一步的进步需要流体力学以及分析、数值和随机数学领域的共同努力。优先方案致力于开发新的数学模型和方法，以了解因双曲非线性而增强或耗尽的小尺度和机制的动态产生。它致力于为双曲输运提供一种新的分析和数值范例，为即将到来的大雷诺数极限下的小尺度湍流理论提供坚实的基础。优先方案将主要围绕三个主要研究方向发展：新的解概念：这包括对流体力学中出现的双曲系统的分析(例如通过广义熵方法、耗散极限或基于概率和矩的解)、为这些解概念设计高分辨率数值，以及探索与现代统计湍流建模和扰动/过滤技术的联系。多尺度模型和渐近流型：研究包括发展和分析模型层级(例如，Boltzmann-Euler或统计湍流)及其闭合，以说明渐近流型(例如，极端马赫数)。需要设计保熵和保结构的数值方法，允许在通过误差控制的模型选择遍历层次结构和制度时保持渐近状态。概率模型：这一领域包括流体力学中出现的双曲系统的随机模型的分析、数值和不确定性量化。它包括使用随机变分原理探索统计湍流的概率建模概念，以及探索用于混合扰动/过滤技术的随机/数据驱动工具。不确定性量化的数值方法应考虑到基本模型的双曲线特征的保留。

Nonlinear hyperbolic balance laws are ubiquitous in the modelling of fluidmechanical processes. They enable the development of powerful numerical simulation methods that back decision-making for critical applications such as in-silico air- and spacecraft design or climate change research. However, fundamental questions about distinctive hyperbolic features remain open including the multi-scale interference of shock and shear waves, or the interplay of hyperbolic transport and random environments. The largely unsolved well-posedness problem for multi-dimensional inviscid flow equations is deeply connected to the laws of turbulent fluid motion in the high Reynolds-number limit. Further progress requires a concerted effort of both fluid mechanics and the mathematical fields of analysis, numerics, and stochastics. The Priority Programme is devoted to the development of new mathematical models and methods to understand the dynamic creation of small scales and mechanisms which are either enhanced or depleted by the hyperbolic nonlinearity. It strives at a novel analytical and numerical paradigm for hyperbolic transport that can provide firm grounds for the upcoming theory of small-scale turbulence in the large Reynolds number limit. The Priority Programme will mostly evolve around three major research directions: Novel solution concepts: This includes the analysis for hyperbolic systems arising in fluid mechanics (via e.g. generalized entropy methods, dissipative limits or probabilistic and moment-based solutions), the design of high-resolution numerics for these solution concepts, and exploring the connections to modern statistical turbulence modelling and perturbation/filtering techniques. Multi-scale models and asymptotic regimes: Research includes the development and analysis of model hierarchies (e.g. Boltzmann-Euler or in statistical turbulence) and their closures that account for asymptotic flow regimes (e.g. extreme Mach numbers). Entropy- and structure-preserving numerical methods need to be designed that allow the preservation of asymptotic states while traversing through hierarchies and regimes by error-controlled model selection. Probabilistic models: This area comprises the analysis, numerics and uncertainty quantification for stochastic models of hyperbolic systems arising in fluid mechanics. It includes probabilistic modelling concepts to explore statistical turbulence using e.g. stochastic variational principles and the exploration of stochastic/data-driven tools for hybrid perturbation/filtering techniques. Numerical methods of uncertainty quantification should account for the preservation of hyperbolic features of the underlying model.