Regularity properties of solutions to the 3D Navier-Stokes equations

切斯基多夫DMS-1517583研究人员研究了有关液体或气体等流体运动方程的几个基本未决问题。这些方程可以解释和预测心血管系统中海洋、大气或血液的循环模式；预测天气变化；描述船只或飞机的运动。它们是在近两个世纪前被引入的，但在数学上仍然没有得到很好的理解。尽管这些方程被物理学家和工程师广泛用于实际应用，但解的存在和唯一性仍然是未知的。解的存在性的数学证明将最终证明这些方程是正确的。失去解的唯一性的数学证明将识别方程的局限性。预计该项目还将阐明与湍流相关的某些基本问题。湍流有时被称为经典物理学中最后一个未解决的问题，它是许多流体流动中的一种关键现象，例如飞机机身、车辆、船舶和涡轮机叶片周围的流体。对湍流的更好的数学理解将导致这些物体设计的改进。80年前，Leray提出湍流是由于流动中奇点的形成，并引入了三维Navier-Stokes方程弱解的概念。这个概念成为偏微分方程组理论中的一个基本概念。本文的目的是研究三维N-S方程弱解的正则性及其与湍流的关系。这项工作的一个方向是构造具有各种异常行为的弱解，例如范数膨胀、不连续或范数不连续。这种现象可能是来自右翼的瞬间爆炸的结果，这比平稳解决方案的爆炸更容易控制。当方程中的强迫项较大时，三维Navier-Stokes方程的解的长时间行为变得非常复杂并且往往是混沌的。对于流体流动，这种现象称为湍流。即使湍流中的速度场是混沌的，实验和数值证据表明，平均速度仍然表现出一定的规律性结构。利用速度矢量场的Littlewood-Paley分解和原子分解，研究了Navier-Stokes方程的Leray-Hopf弱解，证明了湍流的一些经验定律。

CheskidovDMS-1517583 The investigator studies several fundamental open questions concerning the equations governing the motion of fluids, such as liquids or gases. These equations can explain and predict the pattern of circulation of the oceans, atmosphere, or blood in the cardiovascular system; predict changes in weather; describe the motion of a boat or an aircraft. They were introduced almost two centuries ago but are still not well understood mathematically. Even though the equations are broadly used by physicists and engineers for real-life applications, the existence and uniqueness of solutions are still not known. A mathematical proof of existence of solutions would justify the equations definitively. A mathematical proof of a loss of uniqueness of solutions would identify limitations of the equations. The project is also expected to shed light on certain fundamental issues related to turbulence. Turbulence, sometimes referred to as the last unsolved problem in classical physics, is a crucial phenomenon occurring in many fluid flows, for instance those around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence would lead to improvements in the design of these objects. Eighty years ago, Leray suggested that turbulence was due to formation of singularities in the flow and introduced a notion of weak solutions to the three-dimensional Navier-Stokes equations. The notion became a basic concept in the theory of partial differential equations. The aim of this proposal is to study regularity properties of weak solutions to the three-dimensional Navier-Stokes equations and their relation to turbulence. One direction of this work is to construct weak solutions with various abnormal behaviors, such as norm inflation, discontinuity, or norm discontinuity. Such phenomena may be a result of an instantaneous blow-up from the right, which is much easier to control than a blow-up of a smooth solution. When the forcing term in the equations is large, the long-time behavior of solutions to the three-dimensional Navier-Stokes equations becomes very complicated and often chaotic. For fluid flows this phenomenon is known as turbulence. Even when a velocity field in a turbulent flow is chaotic, experimental and numerical evidence show that the averaged velocity still displays some regular structure. The investigator studies Leray-Hopf weak solutions to the Navier-Stokes equations and proves some of the empirical laws of turbulence, using the Littlewood-Paley and atomic decompositions of the velocity vector field.