Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Navier-Stokes equations, determining forms, determining modes, inertial
  manifolds, dissipative dynamical systems   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 continuous
functions of the variable $s\in\mathbb{R}$ with values in certain
finite-dimensional linear space. This new evolution ODE, named {\it 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 projections 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.

24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph