A DETERMINING FORM FOR THE 2D NAVIER-STOKES EQUATIONS - THE FOURIER MODES CASE
A DETERMINING FORM FOR THE 2D NAVIER-STOKES EQUATIONS - THE FOURIER MODES CASE
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发表时间：
2012
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通讯作者：
C. Foias;M. Jolly;R. Kravchenko;E. Titi
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作者：
C. Foias;M. Jolly;R. Kravchenko;E. Titi
.本文证明了二维不可压Navier-Stokes方程（NSE）的决定模满足Banach空间X中变量s ∈ R的所有有界连续函数的dv/dt = F（v）形式的常微分方程。这个新的演化常微分方程被命名为决定形式，它在空间X中导出了一个无限维动力系统，值得注意的原因有两个。一个是F是从X到自身的全局Lipschitz。另一个是决定形式的长期动力学包含NSE的长期动力学;决定形式的行波解，即，形式为v（t，s）= v0（t + s）的那些精确地对应于初始数据v0，该初始数据是NSE的全局吸引子的解在确定模式上的投影。确定的形式也被证明是耗散的，一个吸收球的半径的估计是根据确定模式的数量和Grashof数（一个无量纲的物理参数）。最后，一个统一的艾德的方法概述了一个ODE萨蒂斯艾德的各种其他确定参数，如节点值，有限体积，有限元素。
. The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form dv/dt = F ( v ), in the Banach space, X , of all bounded con- tinuous functions of the variable s ∈ R with values in certain finite-dimensional linear space. This new evolution ODE, named determining form , induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form v ( t, s ) = v 0 ( t + s ), correspond exactly to initial data v 0 that are pro- jections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter). Finally, a unified approach is outlined for an ODE satisfied by a variety of other determining parameters such as nodal values, finite volumes, and finite elements.