Stability of Shock Waves under Hyperbolic Dissipation

该项目研究可压缩流体动力学的PDE系统，该系统不像经典的Navier-Stokes-Fourier （NSF）描述的那样以抛物线方式模拟粘度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。可能的非唯一性问题，可想象的野解，可能的湍流，以及希望消失的粘度极限，这些问题都出现在这些模型中，就像经典模型一样。然而，它们出现在不同的双曲线尺度上。为了预先检验这些尺度，该项目举例调查了双曲-双曲模型中原型冲击波的稳定性，旨在(a)对双曲-抛物线设置中的冲击波的理解与目前现有的理论一样好，(b)描述从双曲-双曲尺度到经典双曲-抛物线尺度的极限。主要的重点是二阶系统，它完全由质量、动量和能量的五个守恒定律组成，并通过使用状态变量的二阶空间和时间导数的算子结合粘度和热传导。近年来，文献主要在相对论背景下讨论了这些模型。该项目的一小部分处理（非相对论性）一阶双曲模型，其中耗散由附加场的放松平衡定律描述，本着扩展热力学的精神。从激波稳定性的角度来看，我们计划研究的标度极限是无限光速的标度极限，即从二阶双曲-双曲到经典可压缩NSF，以及松弛时间消失的标度极限，即从一阶双曲-双曲到经典可压缩NSF。

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.