114 research outputs found
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
Determining asymptotic behavior from the dynamics on attracting sets
Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar results as in the theory of inertial manifolds. Furthermore, such sets are shown to be retracts of the
phase space, which implies that they are simply connected.Ministerio de Educación y CienciaDepartamento de Ecuaciones Diferenciales y Análisis Numérico (Universidad de Sevilla
Embedding of global attractors and their dynamics
Using shape theory and the concept of cellularity, we show that if is the
global attractor associated with a dissipative partial differential equation in
a real Hilbert space and the set has finite Assouad dimension ,
then there is an ordinary differential equation in , with , that has unique solutions and reproduces the dynamics on . Moreover,
the dynamical system generated by this new ordinary differential equation has a
global attractor arbitrarily close to , where is a homeomorphism
from into
Persistence of invariant manifolds for nonlinear PDEs
We prove that under certain stability and smoothing properties of the
semi-groups generated by the partial differential equations that we consider,
manifolds left invariant by these flows persist under perturbation. In
particular, we extend well known finite-dimensional results to the setting of
an infinite-dimensional Hilbert manifold with a semi-group that leaves a
submanifold invariant. We then study the persistence of global unstable
manifolds of hyperbolic fixed-points, and as an application consider the
two-dimensional Navier-Stokes equation under a fully discrete approximation.
Finally, we apply our theory to the persistence of inertial manifolds for those
PDEs which possess them. teComment: LaTeX2E, 32 pages, to appear in Studies in Applied Mathematic
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