Mathematical Sciences: "Mathematical Topics Related to Fluid Instabilities

9622563 Friedlander S.Friedlander将与M.M.Vishik合作，继续她对流体运动中出现的数学问题的研究。提出了研究理想流体关于给定定常流动的小振荡的谱：这导致了对退化的非椭圆微分算子的谱的研究。不稳定连续谱的确切位置将纯粹根据动力系统的量来确定。我们将从解析和数值两方面研究由于光谱的离散部分而导致的流体不稳定的例子。最近的研究证明，在一定的假设条件下，定常无粘流的线性不稳定性意味着非线性(Lyapunov)不稳定性。另一个研究方向是研究控制磁流体力学的增广流体方程组的不稳定性。用局部偏微分方程组的形式，得到了系统不稳定的充分条件。有人建议用这个判据来证明某些天体物理模型的不稳定性。所有流体的运动都会不断地受到微小的干扰(例如，想象一下实验室里的一个水箱被外面开过的卡车“干扰”了)。一个自然的问题是，扰动的影响是否消失了，使流体保持与以前相同的状态，这称为稳定，或者扰动的影响改变了流体的构型，这称为不稳定。一个多世纪以来，流体稳定性/不稳定性问题一直是科学文献中备受关注的经典问题，它是气象学、海洋学、地球物理学和天体物理学研究的基础，尤其是海/气界面的不稳定性对任何全球环境变化的研究都具有重要意义。流体不稳定性的数学是由一个偏微分方程组控制的，这些偏微分方程组非常具有挑战性和趣味性。尽管进行了一个世纪的研究，但仍有许多与这些方程有关的数学问题。该提案将继续解决其中一些悬而未决的问题。***

9622563 Friedlander S.Friedlander in collaboration with M.M.Vishik will continue her investigation into mathematical problems that arise in fluid motion. It is proposed to study the spectrum of small oscillations of an ideal fluid about a given steady flow: this leads to the study of the spectrum of a degenerate non-elliptic differential operator. The exact location of the unstable continuous spectrum will be determined purely in terms of dynamical systems quantities. Examples of fluid instability due to the discrete part of the spectrum will be studied analytically and numerically. It was recently proved that,under certain assumptions, linear instability of a steady inviscid flow implies nonlinear (Lyapunov) instability. It is proposed to extend the range of applicability of this result.Another line of research is the study of instabilities for the augmented system of fluid equations that govern magnetohydrodynamics. A sufficient condition for instability has been derived in terms of a system of local PDE. It is proposed to apply this criterion to demonstrate instability of certain astrophysical models. %%% All fluid motions are continually subject to small disturbances ( eg, think of a tank of water in a laboratory that is "disturbed" by a truck driving by outside ). A natural question arises as to whether the the effect of the disturbance dies away leaving the fluid in the same state as before --this is called stable--or the effect of the disturbance is to change the configuration of the fluid--this is called unstable. The question of fluid stability/instability is a classical one that has received much attention in the scientific literature for more than a century.It is fundamental to studies in meteorology, oceanography, geophysics and astrophysics ; in particular, instabilities at the air/sea interface are significant to any study of the global change of the environment. The mathematics of fluid instabilities is governed by a system of partial differ ential equations that are remarkably challenging and interesting. Despite a century of study there are many open mathematical questions connected with these equations. This proposal will continue to address some of these open problems. ***