303 research outputs found
A determining form for the damped driven Nonlinear Schr\"odinger Equation- Fourier modes case
In this paper we show that the global attractor of the 1D damped, driven,
nonlinear Schr\"odinger equation (NLS) is embedded in the long-time dynamics of
a determining form. The determining form is an ordinary differential equation
in a space of trajectories where is the
-projector onto the span of the first Fourier modes. There is a
one-to-one identification with the trajectories in the global attractor of the
NLS and the steady states of the determining form. We also give an improved
estimate for the number of the determining modes
One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations
The evolution of a determining form for the 2D Navier-Stokes equations (NSE),
which is an ODE on a space of trajectories is completely described. It is
proved that at every stage of its evolution, the solution is a convex
combination of the initial trajectory and the fixed steady state, with a
dynamical convexity parameter , which will be called the characteristic
determining parameter. That is, we show a remarkable separation of variables
formula for the solution of the determining form. Moreover, for a given initial
trajectory, the dynamics of the infinite-dimensional determining form are
equivalent to those of the characteristic determining parameter which
is governed by a one-dimensional ODE. %for the parameter specifying the
position on the line segment. This one-dimensional ODE is used to show that if
the solution to the determining form converges to the fixed state it does so no
faster than , otherwise it converges to a projection
of some other trajectory in the global attractor of the NSE, but no faster than
, as , where is the
evolutionary variable in determining form. The one-dimensional ODE also
exploited in computations which suggest that the one-sided convergence rate
estimates are in fact achieved. The ODE is then modified to accelerate the
convergence to an exponential rate. Remarkably, it is shown that the zeros of
the scalar function that governs the dynamics of , which are called
characteristic determining values, identify in a unique fashion the
trajectories in the global attractor of the 2D NSE. Furthermore, the
one-dimensional characteristic determining form enables us to find
unanticipated geometric features of the global attractor, a subject of future
research
Assimilation of nearly turbulent Rayleigh-B\'enard flow through vorticity or local circulation measurements: a computational study
We introduce a continuous (downscaling) data assimilation algorithm for the
2D B\'enard convection problem using vorticity or local circulation
measurements only. In this algorithm, a nudging term is added to the vorticity
equation to constrain the model. Our numerical results indicate that the
approximate solution of the algorithm is converging to the unknown reference
solution (vorticity and temperature) corresponding to the measurements of the
2D B\'enard convection problem when only spatial coarse-grain measurements of
vorticity are assimilated. Moreover, this convergence is realized using data
which is much more coarse than the resolution needed to satisfy rigorous
analytical estimates
Continuous data assimilation with blurred-in-time measurements of the surface quasi-geostrophic equation
An intrinsic property of almost any physical measuring device is that it
makes observations which are slightly blurred in time. We consider a
nudging-based approach for data assimilation that constructs an approximate
solution based on a feedback control mechanism that is designed to account for
observations that have been blurred by a moving time average. Analysis of this
nudging model in the context of the subcritical surface quasi-geostrophic
equation shows, provided the time-averaging window is sufficiently small and
the resolution of the observations sufficiently fine, that the approximating
solution converges exponentially fast to the observed solution over time. In
particular, we demonstrate that observational data with a small blur in time
possess no significant obstructions to data assimilation provided that the
nudging properly takes the time averaging into account. Two key ingredients in
our analysis are additional boundedness properties for the relevant interpolant
observation operators and a non-local Gronwall inequality.Comment: 44 page
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