Mathematical Sciences: Mathematical Problems in Incompressible Fluid Dynamics

DMS-9501062 PI：Constantin建议在边界驱动的N-S湍流中研究体积耗散的变分界，特别要注意瑞利-贝纳德对流。为了研究结构函数的标度，并将耗散尺度和惯性尺度上的Kolmogorov唯象联系起来，建议对Navier-Stokes湍流不变测度的性质进行解析研究。本文还建议研究欧拉方程和活动标量中的无粘性爆裂，重点是大气流动的准地转模式。其目的是理解能量小尺度的惯性超指数生成。目前的建议是为了理解不可压缩湍流。这类在人类尺度上最常见的物理现象对于从天气预报到计算机技术的大量科学和工业活动都很重要。这一建议针对的是交互工作的数学方面：PI与理论物理学家、实验物理学家、工程师和计算科学家合作，在相同的特定实验情况下工作，并研究相同的理论问题。

DMS-9501062 PI: Constantin It is proposed to work on variational bounds for bulk dissipation in boundary driven Navier-Stokes turbulence, with particular attention to Rayleigh-Benard convection. It is proposed to investigate analytically properties of invariant measures for Navier-Stokes turbulence, with the aim of studying scaling of structure functions and making connection with aspects of the Kolmogorov phenomenology at the dissipative and inertial scales. It is also proposed to investigate inviscid blow-up in the Euler equations and active scalars, with emphasis on the quasi- geostrophic model of atmospheric flow. The aim is to understand inertial super-exponential generation of energetic small scales. The present proposal is aimed at the understanding of incompressible turbulence. This most common class of physical phenomena at the human scale is important for a large number of scientific and industrial activities, from weather prediction to computer technology. This proposal is aimed at the mathematical aspect of an interactive effort: the PI collaborates with theoretical physicists, experimental physicists, engineers and computational scientists, working on the same specific experimental situations and investigating the same theoretical issues.