Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear stability results for plane Couette and Poiseuille flows   In this article we prove, choosing an appropriately weighted $L_2$-energy
equivalent to the classical energy, that the plane Couette and Poiseuille flows
are nonlinearly stable with respect to streamwise perturbations for any
Reynolds number. In this case the coefficient of time-decay of the energy is
$\pi^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of
streamwise perturbations of linearized equations. We also prove that the plane
Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is
less then ${\rm Re}_{Orr}/\sin \varphi$ when the perturbation is a tilted
perturbation, i.e. 2D perturbations with wave vector which forms an angle
$\varphi \in [0, \pi/2]$ with the direction ${\bf i}$ of the motion. ${\rm
Re}_{Orr}$ is the Orr (1907) critical Reynolds number for spanwise
perturbations which, for the Couette flow is ${\rm Re}_{Orr}=177.22$ and for
the Poiseuille flow is ${\rm Re}_{Orr}=175.31$.
  In particular these results improve those obtained by Joseph (1966), who
found for streamwise perturbations a critical nonlinear value of $82.6$ in the
Couette case, and those obtained by Joseph and Carmi (1969), who found the
value $99.1$ for plane Poiseuille flow for streamwise perturbations. The
results we obtain here are, for any angle, in a good agreement with the
experiments (see Prigent et al. 2003) and the numerical simulations (see
Barckley and Tuckerman 2005, 2007).

5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph