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aries1988 : fluid   13

改写数学历史的百万美元大奖
具体的,克雷研究所把这个百万美元大奖问题分为两个问题:

初值问题(柯西问题):如果给定初始的速度和压力的解是有限的,证明存在或者不存在有限的速度场和压力场(且整体的能量是有界的)符合NS方程;

周期问题:如果给定初始的速度和压力的解是周期的,证明存在或者不存在周期的速度场和压力场(且整体的能量是有界的)符合NS方程;

两个问题的解还要保证空间能量的有界。

最通俗的解释就是:对于一个没有任何障碍物、突然改变的能量等突发事件,是否存在一个光滑的解。

更通俗但不严谨的解释就是:数学家们无法求解这个方程。

2007年,陶哲轩写了一篇文章名为“Why global regularity for Navier-Stokes is hard”的博客。在这篇文章中,他认为求解类似NS方程这种的偏微分方程组就像寻找一个大尺度的定量的可以模化的特性,来对不可以预测的行为进行限定。

然而在NS方程描述的流体运动中,并不存在这样一个大尺度的变量。唯一存在的就是流体的全部动能,以及流体内由于摩擦消耗的少部分能量。虽然这些使我们明确知道的,但是这并不是我们理解NS方程的建设性方向。

如果NS方程真的要被求解,要么发现这样一种大尺度的变量来控制全局的能量防止blow-up,要么我们需要一种全新的数学方法来理解。
fluid  mechanics  explained  maths  research 
december 2017 by aries1988
V0088: Turning on a Dime – Asymmetric Vortex Formation in Hummingbird Maneuvering
Hummingbirds are versatile natural flyers that can perform locomotion as insects, such as hovering, forward/backward flight, turning maneuver and more. The unsteady vortex dynamics is key to understand aerodynamic features of these motions. Here we present an integrated approach combining high-speed photogrammetry, wing/body surface tracking, and immersed boundary method based flow simulations to study the three-dimensional vortex dynamics of a freely maneuvering hummingbird. The simulation results of the hummingbird performing pure yaw turn show asymmetric wake structures between the inner and outer wings. Dual-loop vortex structures have been observed in the near wake of the outer wing during downstroke, and of the inner wing during upstroke. The interactions between the wings and these complex vortex structures have implied both aerodynamic and dynamic benefits of the flapping wings in hummingbird’s maneuvering flight. (This work is supported by NSF CBET-1313217 and AFOSR FA9550-12-1-0071)
bird  cfd  fluid  visualization 
january 2016 by aries1988
The Singular Mind of Terry Tao
in the process, he will have also solved the Navier-­Stokes global regularity problem, which has become, since it emerged more than a century ago, one of the most important in all of mathematics.

‘‘A very central part of any mathematician’s life is this sense of connection to other minds, alive today and going back to Pythagoras,’’ said Steven Strogatz, a professor of mathematics at Cornell University. ‘‘We are having this conversation with each other going over the millennia.’’

Long ago, mathematicians invented a number that when multiplied by itself equals negative 1, an idea that seemed to break the basic rules of multiplication. It was so far outside what mathematicians were doing at the time that they called it ‘‘imaginary.’’ Yet imaginary numbers proved a powerful invention, and modern physics and engineering could not function without them.
maths  idea  fluid 
august 2015 by aries1988

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