Mathematical Sciences: "Mathematical Topics Related to Fluid Instabilities

小行星9622563 弗里德兰德在合作MM维希克将继续她的调查数学问题中出现的流体运动。 它建议研究一个理想流体的小振荡的频谱关于一个给定的稳定流：这导致研究的频谱退化的非椭圆型微分算子。不稳定连续谱的确切位置将完全根据动力系统的量来确定。由于频谱的离散部分的流体不稳定的例子将进行分析和数值研究。 最近证明，在某些假设下，定常无粘流的线性不稳定性隐含着非线性（李雅普诺夫）不稳定性。另一个研究方向是研究广义系统的不稳定性， 控制磁流体力学的流体方程。利用局部偏微分方程组导出了不稳定性的一个充分条件。建议应用这一判据来证明某些天体物理模型的不稳定性。 %%% 所有的流体运动都不断地受到微小的扰动（例如，想象一下实验室里的一罐水被外面驶过的卡车“扰动”）。一个很自然的问题出现了，到底是扰动的影响消失了，使流体处于和以前一样的状态--这叫做稳定--还是扰动的影响改变了流体的构型--这叫做不稳定。流体的稳定性/不稳定性是一个经典的问题，在科学文献中受到了广泛的关注超过世纪。它是气象学，海洋学，天体物理学和天体物理学研究的基础;特别是，在空气/海洋界面的不稳定性是重要的任何研究的全球环境变化。流体不稳定性的数学是由一个偏微分方程系统，这是非常具有挑战性和有趣的。尽管经过了世纪的研究，但仍有许多与这些方程有关的开放式数学问题。这项建议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. ***