Small Scales in the Navier-Stokes Equations

这个项目将解决在粘性不可压缩流动中产生的各种自然小尺度的定义、相互作用和严格估计。特别是，我们将考虑那些可以直接从Navier-Stokes方程的解中获得的长度，该方程是流体流动的主要模型。这种尺度的例子是由解的傅里叶谱产生的那些尺度，例如傅里叶谱被指数截断的波数的倒数，以及测量解的水平集的复杂性的长度尺度。我们还将讨论流体力学中出现的其他耗散偏微分方程组的相关问题，如Ginzburg-Landau方程和Kuramoto-Sivashinsky方程。N-S方程可能是研究最广泛的非线性偏微分方程组。它们通常被认为含有解释大部分湍流现象的必要成分。这个项目将解决解决方案的性质，这将对理解流动中精细结构的产生和性质具有重要意义。潜在的应用包括对流体流动的复杂性进行量化、建立解的光谱特性、估计数值求解流动所需的网格大小、以及关于为流动监测定位可观测对象的信息

0072662KukavicaThis project will address definition, interplay, and rigorous estimates of various natural small scales arising in a viscous incompressible flow. In particular, we will consider those lengths that can be derived directly from solutions of the Navier-Stokes equations, which is the main model for a fluid flow. Examples of such scales are the those resulting from the Fourier spectrum of solutions, such as the inverse of the wave number at which the Fourier spectrum is cut off exponentially, and the length scales measuring complexity of level sets of solutions. Related problems will be addressed also for other dissipative partial differential equations arising in fluid dynamics, such as the Ginzburg-Landau and the Kuramoto-Sivashinsky equations.Navier-Stokes equations are perhaps the most widely studied system of nonlinear partial differential equations. They are generally believed to contain necessary ingredients to explain much of turbulence phenomena. This project will address properties of solutions which would have implications in understanding creation and properties of fine structures in a flow. Potential applications include quantifying complexity of a fluid flow, establishment of spectral properties of solutions, estimation of the size of the mesh needed to resolve a flow numerically, and information on locating observables for the flow's monitoring