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On the controllability and stabilization of the linearized Benjamin-Ono equation

Felipe Linares, Jaime H. Ortega (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law...

On the dimension of the attractor for a perturbed 3d Ladyzhenskaya model

Dalibor Pražák, Josef Žabenský (2013)

Open Mathematics

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ɛΔ3. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ɛ > 0.

On the dynamics of equations with infinite delay

Dalibor Pražák (2006)

Open Mathematics

We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

Periodic solutions of dissipative dynamical systems with singular potential and p-Laplacian

Bing Liu (2002)

Annales Polonici Mathematici

By using the topological degree theory and some analytic methods, we consider the periodic boundary value problem for the singular dissipative dynamical systems with p-Laplacian: ( ϕ p ( x ' ) ) ' + d / d t g r a d F ( x ) + g r a d G ( x ) = e ( t ) , x(0) = x(T), x’(0) = x’(T). Sufficient conditions to guarantee the existence of solutions are obtained under no restriction on the damping forces d/dt gradF(x).

Perturbations of the harmonic map equation

Thomas Kappeler (2002)

Journées équations aux dérivées partielles

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed riemannian manifolds and the latter is in addition of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is...

Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach

Aleksander Ćwiszewski (2011)

Open Mathematics

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions...

Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains

Cung The Anh, Dang Thanh Son (2015)

Annales Polonici Mathematici

We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality with a large class of non-autonomous external forces. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal...

Recent progress in attractors for quintic wave equations

Anton Savostianov, Sergey Zelik (2014)

Mathematica Bohemica

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ℝ 3 with damping terms of the form ( - Δ x ) θ ∂ t u , where θ = 0 or θ = 1 / 2 . The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when θ = 1 / 2 . For θ = 0 existence of smooth attractors is more complicated and follows...

Rigidity results for Bernoulli actions and their von Neumann algebras

Stefaan Vaes (2005/2006)

Séminaire Bourbaki

Using very original methods from operator algebras, Sorin Popa has shown that the orbit structure of the Bernoulli action of a property (T) group, completely remembers the group and the action. This information is even essentially contained in the crossed product von Neumann algebra. This is the first von Neumann strong rigidity theorem in the literature. The same methods allow Popa to obtain II 1 factors with prescribed countable fundamental group.

Selection Theorem for Systems with Inheritance

A. N. Gorban (2010)

Mathematical Modelling of Natural Phenomena

The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural”...

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