Mathematical Sciences: Regularity and Oscillations in Mathematical Theory of an Ideal Incompressible Fluid

DMS-9531769 Vishik The PI将研究流体运动的数学问题。调和分析和偏微分方程组的方法将被用来研究理想流体不可压缩流动的奇点演化和光滑性损失。可用于分析的典型涡量场并不本质上是有界的，尽管它爆炸的集合在某种意义上是小的。这项拟议的研究将提供有关液体颗粒拉格朗日轨迹分离和流动产生小尺度结构的准确信息。PI将与S.Friedlander合作，利用动力系统和谱理论的方法，继续研究理想流体在给定平衡附近的非线性不稳定性和小振荡的频谱。流体运动的数学理论旨在从基本方程中理解流体中的现象，如湍流。这门学科是气象学、地球物理学、天体物理学、海洋学和空气动力学应用的基础。在过去，流体运动的偏微分方程式是数学中几个重要概念的来源。拟议的研究将解决两个问题。首先，将研究当速度场中存在温和奇点时流体流动演变的一种特殊情况。理想流体的这种流动可以在很高的雷诺数下近似于粘性流动。第二个研究方向是流体动力稳定性理论。它的目的是了解流体失去稳定性的流动参数和几何形状的一般条件，并描述当流动离稳定状态不太远时，微小振荡的频率。***

DMS-9531769 Vishik The PI will investigate the mathematical problems of fluid motion. Methods of harmonic analysis and partial differential equations will be used to study the evolution of singularities and loss of smoothness for the incompressible flows of an ideal fluid. The typical vorticity field admissible for the analysis is not essentially bounded although the set where it blows up is in some sense small. The proposed study will provide precise information about the separation of Lagrangian trajectories of liquid particles and generation of small scale structures by the flow. The PI in collaboration with S.Friedlander will continue to investigate nonlinear instability and spectrum of small oscillations of an ideal fluid about a given equilibrium using methods of dynamical systems and spectral theory. %%% Mathematical theory of fluid motion is aimed at understanding phenomena in fluids, such as turbulence, from the basic equations. This subject is fundamental for applications in meteorology, geophysics, astrophysics, oceanology, and aerodynamics.In the past, the partial differential equations of fluid motion have been a source of several important concepts in mathematics. The proposed research will address two problems. First, one particular scenario of the evolution of fluid flow when mild singularities are present in the velocity field, will be investigated. Such flows of an ideal fluid may approximate viscous flows at very high Reynolds number. The second line of research is hydrodynamic stability theory.Its goal is to understand the general conditions for the parameters and geometry of the flow under which fluid loses stability and to describe frequencies of small oscillations when the flow is not too far from the steady state. ***