1,229 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
Selection of quasi-stationary states in the Navier-Stokes equation on the torus
The two dimensional incompressible Navier-Stokes equation on with , periodic boundary
conditions, and viscosity is considered. Bars and dipoles, two
explicitly given quasi-stationary states of the system, evolve on the time
scale and have been shown to play a key role in its
long-time evolution. Of particular interest is the role that plays in
selecting which of these two states is observed. Recent numerical studies
suggest that, after a transient period of rapid decay of the high Fourier
modes, the bar state will be selected if , while the dipole will
be selected if . Our results support this claim and seek to
mathematically formalize it. We consider the system in Fourier space, project
it onto a center manifold consisting of the lowest eight Fourier modes, and use
this as a model to study the selection of bars and dipoles. It is shown for
this ODE model that the value of controls the behavior of the
asymptotic ratio of the low modes, thus determining the likelihood of observing
a bar state or dipole after an initial transient period. Moreover, in our
model, for all , there is an initial time period in which the
high modes decay at the rapid rate , while the low
modes evolve at the slower rate. The results for the
ODE model are proven using energy estimates and invariant manifolds and further
supported by formal asymptotic expansions and numerics.Comment: 29 pages, 4 figure
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 1/ν, which is much faster than what would be produced by the static fluid if its viscosity is 0<ν≪1. 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 1/ν. 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.https://arxiv.org/abs/1804.06916First author draf
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
Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus
The stochastically forced vorticity equation associated with the two
dimensional incompressible Navier-Stokes equation on
is considered for ,
periodic boundary conditions, and viscocity . An explicit family of
quasi-stationary states of the deterministic vorticity equation is known to
play an important role in the long-time evolution of solutions both in the
presence of and without noise. Recent results show the parameter plays
a central role in selecting which of the quasi-stationary states is most
important. In this paper, we aim to develop a finite dimensional model that
captures this selection mechanism for the stochastic vorticity equation. This
is done by projecting the vorticity equation in Fourier space onto a center
manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo
simulation, the vorticity equation and the model are shown to be in agreement
regarding key aspects of the long-time dynamics. Following this comparison,
perturbation analysis is performed on the model via averaging and
homogenization techniques to determine the leading order dynamics for
statistics of interest for .Comment: 23 pages, 27 figure
Nonlinear stability of source defects in oscillatory media
In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined with an instantaneous phase-tracking technique to obtain detailed pointwise estimates describing perturbations to lowest order as a phase-shift radiating outward at a linear rate plus a pair of localized approximately Gaussian excitations along the phase-shift boundaries; we show that in the wake of these outgoing waves the perturbed solution converges time-exponentially to a space-time translate of the original source pattern.https://arxiv.org/abs/1802.07676First author draf
Nonlinear stability of source defects in the complex Ginzburg-Landau equation
In an appropriate moving coordinate frame, source defects are time-periodic
solutions to reaction-diffusion equations that are spatially asymptotic to
spatially periodic wave trains whose group velocities point away from the core
of the defect. In this paper, we rigorously establish nonlinear stability of
spectrally stable source defects in the complex Ginzburg-Landau equation. Due
to the outward transport at the far field, localized perturbations may lead to
a highly non-localized response even on the linear level. To overcome this, we
first investigate in detail the dynamics of the solution to the linearized
equation. This allows us to determine an approximate solution that satisfies
the full equation up to and including quadratic terms in the nonlinearity. This
approximation utilizes the fact that the non-localized phase response,
resulting from the embedded zero eigenvalues, can be captured, to leading
order, by the nonlinear Burgers equation. The analysis is completed by
obtaining detailed estimates for the resolvent kernel and pointwise estimates
for the Green's function, which allow one to close a nonlinear iteration
scheme.Comment: 53 pages, 5 figure
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