Applied Analysis of the Navier-Stokes and Related Equations

这项数学物理和应用数学的基础研究集中于不可压缩的Navier-Stokes方程和相关的流体动力学方程所带来的挑战。Navier-Stokes方程构成了流体流动的基本数学模型，其解被认为包含了湍流动力学。流体力学中的湍流一直是理论物理和应用数学面临的突出挑战之一，在科学和工程的许多领域都有重要的应用。这项工作是由首席研究员和一名正在撰写博士论文的数学研究生通过现代应用和数值分析进行的。该项目有三个具体目标。第一个是将推导湍流流量理论边界的严格技术推广到Marangoni对流中热传输的新应用，更广泛地应用于具有施加应力边界条件的流体系统中的湍流。其次，我们的目标是将背景场方法推广到无边界流动区域，以推导出流经紧凑体的湍流阻力系数的理论极限。例如，确定球体在粘性流体中高速运动时所受阻力的极限仍然是一个悬而未决的问题，这在数学上是严格的，在物理上也是相关的。第三个目标是利用为N-S方程的解的解析推广而导出的一组动力学方程来研究湍流中出现的小长度尺度。这方面的数学结果将在与湍流速度场中傅里叶功率谱的高波数指数衰减相关的小长度尺度上产生严格的下界。

This fundamental research in mathematical physics and applied mathematics focuses on the challenges presented by the incompressible Navier-Stokes and related equations of fluid dynamics. The Navier-Stokes equations constitute the basic mathematical model for fluid flow, and are believed to contain turbulent dynamics among their solutions. Turbulence in fluid mechanics remains one of the outstanding challenges for theoretical physics and applied mathematics with important applications in many fields of science and engineering. The work is carried out via modern applied and numerical analysis by the principal investigator and a mathematics graduate student doing doctoral dissertation work.The project has three specific objectives. The first is to extend a rigorous technique for deriving theoretical bounds on turbulent flow quantities to new applications for heat transport in Marangoni convection, and more generally to turbulence in fluid systems with imposed stress boundary conditions. Second, we aim to extend the background field method for applications to unbounded flow domains to derive theoretical limits on turbulent drag coefficients for flows past a compact body. It remains an open problem, for example, to establish a limit on the drag experienced by a sphere moving at high speed through a viscous fluid which is both mathematically rigorous and physically relevant. The third objective is to investigate small length scales appearing in turbulent flows by means of a set of dynamical equations derived for an analytic extension of solutions of the Navier-Stokes equations. Mathematical results in this area will produce strict lower bounds on the small length scales associated with high wavenumber exponential decay of the Fourier power spectrum in turbulent velocity fields.