José Langa

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Alexander Korsunsky

University of Oxford

Iolanda Craifaleanu

Technical University Of Civil Engineering, Bucharest

John Fox

University of Oxford

Dr. J. M. Ashfaque (MInstP)

Independent Researcher

ANNIE SELDEN

New Mexico State University

William Rodriguez Serquen

Universidad Nacional Pedro Ruiz Gallo

Nasreen Kausar

Yildiz Technical University

Shyam Asolekar

Indian Institute of Technology Bombay

Dag I K Sjoberg

University of Oslo

Armando Marques-Guedes

UNL - New University of Lisbon

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Papers by José Langa

Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations

Communications on Pure and Applied Analysis, 2007

ABSTRACT The existence and finite fractal dimension of a pullback attractor in the space V for a ... more ABSTRACT The existence and finite fractal dimension of a pullback attractor in the space V for a three-dimensional system of the nonautonomous globally modified Navier-Stokes equations on a bounded domain is established under appropriate properties on the time dependent forcing term. These equations have been proposed recently by T. Caraballo, J. Real and P. E. Kloeden [Adv. Nonlinear Stud. 6, No. 3, 411–436 (2006; Zbl 1220.35115)] and are obtained from the Navier-Stokes equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.

A Gradient-like nonautonomous Evolution Process

International Journal of Bifurcation and Chaos, 2010

In this paper we consider a dissipative damped wave equation with non-autonomous damping of the f... more In this paper we consider a dissipative damped wave equation with non-autonomous damping of the form u tt + β(t)u t = ∆u + f (u) (1) in a bounded smooth domain Ω ⊂ R n with Dirichlet boundary conditions, where f is a dissipative smooth nonlinearity and the damping β : R → (0, ∞) is a suitable function. We prove, if (1) has finitely many equilibria, that all global bounded solutions of (1) are backwards and forwards asymptotic to equilibria. Thus, we give a class of examples of non-autonomous evolution processes for which the structure of the pullback attractors is well understood. That complements the results of [Carvalho & Langa, 2009] on characterization of attractors, where it was shown that a small non-autonomous perturbation of an autonomous gradient-like evolution process is also gradient-like. Note that the evolution process associated to (1) is not a small non-autonomous perturbation of any autonomous gradient-like evolution processes. Moreover, we are also able to prove that the pullback attractor for (1) is also a forwards attractor and that the rate of attraction is exponential.

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Stabilisation of differential inclusions and PDEs without uniqueness by noise

Communications on Pure and Applied Analysis, 2008

We prove that the asymptotic behaviour of partial differential inclusions and partial differentia... more We prove that the asymptotic behaviour of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilised by adding some suitable Itô noise as an external perturbation. We show how the theory previously developed for the single-valued cases can be successfully applied to handle these set-valued cases. The theory of random dynamical systems is used as an appropriate tool to solve the problem.

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Random attractors for stochastic 2D-Navier–Stokes equations in some unbounded domains

by Grzegorz Lukaszewicz and José Langa

Journal of Differential Equations, 2013

We show that the stochastic flow generated by the Stochastic Navier-Stokes equations in a 2-dimen... more We show that the stochastic flow generated by the Stochastic Navier-Stokes equations in a 2-dimensional Poincaré domain has a unique random attractor. This result complements a recent result by Brzeźniak and Li [10] who showed that the flow is asymptotically compact and generalizes a recent result by Caraballo et al. who proved existence of a unique pullback attractor for the time-dependent deterministic Navier-Stokes equations in a 2-dimensional Poincaré domain.

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Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations

Communications on Pure and Applied Analysis, 2007