Applied Analysis of the Navier-Stokes and Related Equations

数学物理和应用数学的基础研究重点是不可压缩Navier-Stokes方程和流体动力学相关方程所带来的挑战。 Navier-Stokes方程构成流体流动的基本数学模型，并且被认为在其解中包含湍流动力学。 流体力学中的湍流是理论物理和应用数学面临的一个突出挑战，在许多科学和工程领域有着重要的应用。 这项工作是通过现代应用和数值分析由首席研究员和一名正在做博士论文的数学研究生进行的。该项目有三个具体目标。 第一个是扩展一个严格的技术，推导出理论界限的湍流量的新应用，在Marangoni对流热传输，更一般的湍流施加应力边界条件的流体系统。 其次，我们的目标是扩展的背景场方法的应用程序无界流域推导出理论限制湍流阻力系数的流动通过一个紧凑的机构。 例如，对高速移动通过粘性流体的球体所经历的阻力进行限制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。 第三个目标是调查小的长度尺度出现在湍流中的一组动力学方程的解析扩展的Navier-Stokes方程的解决方案。 在这方面的数学结果将产生严格的下限的小长度尺度与高波数指数衰减的傅立叶功率谱在湍流速度场。

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.