107 research outputs found
Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non translation-compact external forces
We give a comprehensive study of strong uniform attractors of non-autonomous
dissipative systems for the case where the external forces are not translation
compact. We introduce several new classes of external forces which are not
translation compact, but nevertheless allow to verify the attraction in a
strong topology of the phase space and discuss in a more detailed way the class
of so-called normal external forces introduced before. We also develop a
unified approach to verify the asymptotic compactness for such systems based on
the energy method and apply it to a number of equations of mathematical physics
including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains
Inertial manifolds and finite-dimensional reduction for dissipative PDEs
These notes are devoted to the problem of finite-dimensional reduction for
parabolic PDEs. We give a detailed exposition of the classical theory of
inertial manifolds as well as various attempts to generalize it based on the
so-called Man\'e projection theorems. The recent counterexamples which show
that the underlying dynamics may be in a sense infinite-dimensional if the
spectral gap condition is violated as well as the discussion on the most
important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by
the author as a part of the crash course in the Analysis of Nonlinear PDEs at
Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9,
2012
Exponential attractors for random dynamical systems and applications
The paper is devoted to constructing a random exponential attractor for some
classes of stochastic PDE's. We first prove the existence of an exponential
attractor for abstract random dynamical systems and study its dependence on a
parameter and then apply these results to a nonlinear reaction-diffusion system
with a random perturbation. We show, in particular, that the attractors can be
constructed in such a way that the symmetric distance between the attractors
for stochastic and deterministic problems goes to zero with the amplitude of
the random perturbation.Comment: 37 page
Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations
A nonlinear parabolic equation of the fourth order is analyzed. The equation
is characterized by a mobility coefficient that degenerates at 0. Existence of
at least one weak solution is proved by using a regularization procedure and
deducing suitable a-priori estimates. If a viscosity term is added and
additional conditions on the nonlinear terms are assumed, then it is proved
that any weak solution becomes instantaneously strictly positive. This in
particular implies uniqueness for strictly positive times and further
time-regularization properties. The long-time behavior of the problem is also
investigated and the existence of trajectory attractors and, under more
restrictive conditions, of strong global attractors is shown
Global well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space
We prove the global well-posedness of the so-called hyperbolic relaxation of
the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity
of the sub-quintic growth rate. Moreover, the dissipativity and the existence
of a smooth global attractor in the naturally defined energy space is also
verified. The result is crucially based on the Strichartz estimates for the
linear Scroedinger equation in R^3
A note on a strongly damped wave equation with fast growing nonlinearities
A strongly damped wave equation including the displacement depending
nonlinear damping term and nonlinear interaction function is considered. The
main aim of the note is to show that under the standard dissipativity
restrictions on the nonlinearities involved the initial boundary value problem
for the considered equation is globally well-posed in the class of sufficiently
regular solutions and the semigroup generated by the problem possesses a global
attractor in the corresponding phase space. These results are obtained for the
nonlinearities of an arbitrary polynomial growth and without the assumption
that the considered problem has a global Lyapunov function
Infinite energy solutions for Dissipative Euler equations in R^2
We study the Euler equations with the so-called Ekman damping in the whole 2D
space. The global well-posedness and dissipativity for the weak infinite energy
solutions of this problem in the uniformly local spaces is verified based on
the further development of the weighted energy theory for the Navier-Stokes and
Euler type problems. In addition, the existence of weak locally compact global
attractor is proved and some extra compactness of this attractor is obtained.Comment: arXiv admin note: text overlap with arXiv:1203.573
- …