457 research outputs found
Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
Quasi-stationary, or metastable, states play an important role in
two-dimensional turbulent fluid flows where they often emerge on time-scales
much shorter than the viscous time scale, and then dominate the dynamics for
very long time intervals. In this paper we propose a dynamical systems
explanation of the metastability of an explicit family of solutions, referred
to as bar states, of the two-dimensional incompressible Navier-Stokes equation
on the torus. These states are physically relevant because they are associated
with certain maximum entropy solutions of the Euler equations, and they have
been observed as one type of metastable state in numerical studies of
two-dimensional turbulence. For small viscosity (high Reynolds number), these
states are quasi-stationary in the sense that they decay on the slow, viscous
timescale. Linearization about these states leads to a time-dependent operator.
We show that if we approximate this operator by dropping a higher-order,
non-local term, it produces a decay rate much faster than the viscous decay
rate. We also provide numerical evidence that the same result holds for the
full linear operator, and that our theoretical results give the optimal decay
rate in this setting.Comment: 21 pages, 2 figures. Version 3: minor error from version 2 correcte
Breathers as Metastable States for the Discrete NLS equation
We study metastable motions in weakly damped Hamiltonian systems. These are
believed to inhibit the transport of energy through Hamiltonian, or nearly
Hamiltonian, systems with many degrees of freedom. We investigate this question
in a very simple model in which the breather solutions that are thought to be
responsible for the metastable states can be computed perturbatively to an
arbitrary order. Then, using a modulation hypothesis, we derive estimates for
the rate at which the system drifts along this manifold of periodic orbits and
verify the optimality of our estimates numerically.Comment: Corrected typos. Added Acknowledgmen
Using global invariant manifolds to understand metastability in Burgers equation with small viscosity
The large-time behavior of solutions to Burgers equation with small viscosity
is described using invariant manifolds. In particular, a geometric explanation
is provided for a phenomenon known as metastability, which in the present
context means that solutions spend a very long time near the family of
solutions known as diffusive N-waves before finally converging to a stable
self-similar diffusion wave. More precisely, it is shown that in terms of
similarity, or scaling, variables in an algebraically weighted space, the
self-similar diffusion waves correspond to a one-dimensional global center
manifold of stationary solutions. Through each of these fixed points there
exists a one-dimensional, global, attractive, invariant manifold corresponding
to the diffusive N-waves. Thus, metastability corresponds to a fast transient
in which solutions approach this "metastable" manifold of diffusive N-waves,
followed by a slow decay along this manifold, and, finally, convergence to the
self-similar diffusion wave
Rigorous justification of Taylor dispersion via center manifolds and hypocoercivity
Taylor diffusion (or dispersion) refers to a phenomenon discovered
experimentally by Taylor in the 1950s where a solute dropped into a pipe with a
background shear flow experiences diffusion at a rate proportional to ,
which is much faster than what would be produced by the static fluid if its
viscosity is . This phenomenon is analyzed rigorously using the
linear PDE governing the evolution of the solute. It is shown that the solution
can be split into two pieces, an approximate solution and a remainder term. The
approximate solution is governed by an infinite-dimensional system of ODEs that
possesses a finite-dimensional center manifold, on which the dynamics
correspond to diffusion at a rate proportional to . The remainder term
is shown to decay at a rate that is much faster than the leading order behavior
of the approximate solution. This is proven using a spectral decomposition in
Fourier space and a hypocoercive estimate to control the intermediate Fourier
modes.Comment: 37 pages, 0 figure
Higher Order Modulation Equations for a Boussinesq Equation
In order to investigate corrections to the common KdV approximation to long
waves, we derive modulation equations for the evolution of long wavelength
initial data for a Boussinesq equation. The equations governing the corrections
to the KdV approximation are explicitly solvable and we prove estimates showing
that they do indeed give a significantly better approximation than the KdV
equation alone. We also present the results of numerical experiments which show
that the error estimates we derive are essentially optimal
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