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

DMS-9531769 Visik PI将研究流体运动的数学问题。 调和分析和偏微分方程的方法将 用于研究奇点的演化和光滑性的丧失 对于理想流体的不可压缩流动。典型涡度 分析的容许域不是本质上有界的，尽管它爆破的集合在某种意义上是小的。拟定研究 将提供关于拉格朗日分离的精确信息 液体颗粒的轨迹和由流动产生的小尺度结构。PI与S.Friedlander合作将继续 用动力系统和谱理论方法研究理想流体关于给定平衡点的非线性不稳定性和小振荡的谱。 %%% 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，例如湍流。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。在过去，偏微分 流体运动方程是几个重要概念的来源 在数学上。拟议的研究将解决两个问题。第一、 一种特殊的情况下，当温和的流体流动的演变， 奇点是目前在速度场，将进行调查。 理想流体的这种流动可以在非常高的温度下近似于粘性流动。 雷诺数第二个研究方向是水动力稳定性 其目标是了解参数的一般条件 以及流体失去稳定性的流动的几何形状，并描述 小振荡的频率时，流是不是太远， 稳态 ***

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. ***