Stability of Shock Waves under Hyperbolic Dissipation

双曲线耗散下冲击波的稳定性

• 批准号: 526003069
• 负责人: Professor Dr. Heinrich Freistühler
• 金额: --
• 依托单位: Department of Mathematics
• 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/526003069?language=en
• 关键词: Stability; Shock; Waves; under; Hyperbolic

This project studies PDE systems of compressible fluid dynamics that model the dissipative mechanisms of viscosity and heat conduction not parabolically as the classical Navier-Stokes-Fourier (NSF) description, but, like the non-dissipative part of the evolution, through hyperbolic operators. The issues of possible non-uniqueness, conceivable wild solutions, likely turbulence, and hoped for vanishing-viscosity limit arise for these models just as for the classical one. However, they appear on different, hyperbolic scales. To preexamine these scales, the project exemplarily investigates the stability of prototypical shock waves in hyperbolic-hyperbolic models, aiming at (a) an understanding as good as the presently existing theory for shock waves in the hyperbolic-parabolic setting and (b) a description of limits from the hyperbolic-hyperbolic scales to the classical hyperbolic-parabolic scale. The main emphasis is on second-order systems that consist exclusively of the five conservation laws for mass, momentum and energy, and incorporate viscosity and heat conduction through operators using second-order space and time derivatives of the state variables. In recent years, the literature has discussed such models notably in the relativistic setting. A smaller part of the project deals with (non-relativistic) hyperbolic models of first order in which the dissipation is described by relaxing balance laws for additional fields, in the spirit of Extended Thermodynamics. The scaling limits we plan to study from the point of view of shock wave stability are that of infinite light speed, i.e., from second-order hyperbolic-hyperbolic to classical compressible NSF, and that of vanishing relaxation times, i.e., from first-order hyperbolic-hyperbolic to classical compressible NSF.

该项目研究可压缩流体动力学的PDE系统，该PDE系统不像经典的Navier-Stokes-Fourier（NSF）描述那样抛物线地模拟粘性和热传导的耗散机制，而是像演化的非耗散部分一样，通过双曲算子。可能的非唯一性，可想象的野生解决方案，可能的湍流，并希望消失的粘度极限的问题出现在这些模型中，就像经典的。然而，它们出现在不同的双曲线尺度上。为了预先检查这些尺度，该项目示例性地研究了双曲-双曲模型中原型冲击波的稳定性，目的是（a）与双曲-抛物线设置中的冲击波的现有理论一样好的理解和（B）描述从双曲-双曲尺度到经典双曲-抛物线尺度的限制。主要重点是二阶系统，包括专门的五个守恒定律的质量，动量和能量，并纳入粘度和热传导通过运营商使用二阶空间和时间导数的状态变量。近年来，文献讨论了这样的模型，特别是在相对论设置。该项目的一小部分涉及（非相对论性）双曲模型的一阶，其中耗散是通过放松平衡定律的额外领域，在扩展热力学的精神。我们计划从激波稳定性的角度研究的尺度极限是无限光速，即，从二阶双曲-双曲到经典可压缩NSF，以及弛豫时间消失，即，从一阶双曲-双曲到经典可压缩NSF。