Convex integration and dissipative anomaly in compressible turbulence

高雷诺数流体湍流中统计性质在小尺度上的普适性的基本方面主要是针对不可压缩情况进行的研究，而对于完全可压缩情况的类似研究直到最近才开始。造成这种情况的一个原因是，在（泰勒微尺度）雷诺数R_lambda旁边必须包含进一步的无量纲参数。最近对直接数值模拟（DNS）数据的综合记录的比较表明，不仅必须包括R_lambda旁边的湍流马赫数M_t，而且还必须包括膨胀速度与螺线线均方根速度（rms）的比率delta。另一方面，一个尺度上的能量平衡，类似于Duchon和Robert在不可压缩情况下的工作，揭示了两种不同的可能的级联机制，称为变形和气压功，这可能是造成可压缩湍流耗散异常的原因。本建议的基本前提是，进一步研究可压缩湍流需要详细研究这些机制及其对M_t和δ的依赖关系。虽然变形工作与不可压缩情况和Onsager的经典猜想密切相关，并且预计在非弹性状态下占主导地位，但膨胀主导状态迄今为止完全未被探索。本项目的目的是将现有的onsager弱解理论统一地推广到整个m_t δ参数平面上的可压缩欧拉方程组。数学方面的关键里程碑问题包括可容许弱解的一般非唯一性、onsager临界Besov空间中弱解的构造以及极端耗散事件几何理论的数学基础。流体力学方面的关键里程碑问题是对充分发展的可压缩湍流中与能量耗散事件有关的变形和气压功的尺度相关数值分析。因此，达到这些科学目标需要数学和基础流体力学研究的共同努力，将分析计算、凸积分的建设性证明技术、直接数值模拟以及可压缩湍流的统计和几何分析结合起来。我们的项目特别针对优先计划的研究领域A，侧重于弱熵解的分析和数值研究及其与可压缩纳维-斯托克斯湍流中小尺度间歇性的相关性。

Fundamental aspects of the universality of statistical properties at the small scales in high-Reynolds-number fluid turbulence have been mostly studied for the incompressible case and similar efforts for the fully compressible regime have been started only recently. One reason for this circumstance is that further dimensionless parameters have to be included next to the (Taylor microscale) Reynolds number R_lambda. A recent comparison of a comprehensive record of direct numerical simulation (DNS) data suggests that not only the turbulent Mach number M_t next to R_lambda, but also the ratio of dilatational to solenoidal root mean square (rms) velocities, delta, have to be included. On the other hand, a scale-by-scale energy balance, analogous to work of Duchon and Robert for the incompressible case, reveals two distinct possible cascade mechanisms called deformation and baropycnal work, which may be responsible for the dissipative anomaly in compressible turbulence. The basic premise of this proposal is that further progress in compressible turbulence requires the study of these mechanisms in detail and their dependencies on M_t and delta. Whilst deformation work is closely related to the incompressible case and the classical conjecture of Onsager and is expected to dominate in the anelastic regime, the dilatation-dominated regime is so far completely unexplored. The present project aims at extending the currently available Onsager-theory of weak solutions to the compressible Euler equation system in the entire M_t—delta parameter plane in a unifying manner. Key milestone problems on the mathematical side include the generic non-uniqueness of admissible weak solutions, the construction of weak solutions in Onsager-critical Besov spaces as well as the mathematical foundations of a geometric theory of extreme dissipation events. The key milestone problem on the fluid mechanical side is the scale-dependent numerical analysis of deformation and baropycnal work in relation with energy dissipation events in fully developed compressible turbulence. Reaching these scientific objectives requires consequently a joint effort of mathematical and basic fluid mechanical investigations that brings together analytical calculations, constructive proof techniques of convex integration, direct numerical simulations, and statistical and geometrical analysis for compressible turbulence. Our project addresses in particular the research area A of the Priority Programme, focusing on the analytical and numerical study of weak entropy solutions and their relevance for the small-scale intermittency in compressible Navier-Stokes turbulence.