Pedro Marín-Rubio

A Comparison between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball (1997) to treat equations whose solutions may not be unique, and that due to Melnik and Valero (1998) tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Balls theory to differential inclusions.

Attractors for The Stochastic 3D Navier-Stokes Equations

In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3D Navier–Stokes equations have a global attractor provided that all weak solutions are continuous from (0, ∞) into L2. In this paper we adapt his framework to treat stochastic equations: we introduce a notion of a stochastic generalised semiflow, and then show a similar result to Balls concerning the attractor of the stochastic 3D Navier–Stokes equations with additive white noise.

Pullback Attractors for 2D Navier-Stokes Equations with Delays and Their Regularity

Abstract In this paper we obtain some results on the existence of solution, and of pullback attractors, for a 2D Navier-Stokes model with finite delay studied in [4] and [6]. Actually, we prove a result of existence and uniqueness of solution under less restrictive assumptions than in [4]. More precisely, we remove a condition on square integrable control of the memory terms, which allows us to consider a bigger class of delay terms (for instance, just under a measurability condition on the delay function leading the delayed time). After that, we deal with dynamical systems in suitable phase spaces within two metrics, the L2 norm and the H1 norm. Moreover, we prove that under these assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Finally, from comparison results of attractors we establish relations among them, and under suitable additional assumptions we conclude that these families of attractors are in fact the same object.

Weak Pullback Attractors of Non-autonomous Difference Inclusions

Weak pullback attractors are defined for non-autonomous difference inclusions and their existence and upper semi continuous convergence under perturbation is established. Unlike strong pullback attractors, invariance and pullback attraction here are required only for (at least) a single trajectory rather than all trajectories at each starting point. The concept is thus useful, in particular, for discrete time control systems.

On the Convergence of Solutions of Globally Modified Navier-Stokes Equations with Delays to Solutions of Navier-Stokes Equations with Delays

Abstract We prove that under suitable assumptions, from a sequence of solutions of Globally Modified Navier-Stokes equations with delays we can extract a subsequence which converges in an adequate sense to a weak solution of a three-dimensional Navier-Stokes equation with delays. An additional case with a family of different delays involved in the approximating problems is also discussed.

Some Results on Stochastic Differential Equations with Reflecting Boundary Conditions

Some results related to stochastic differential equations with reflecting boundary conditions (SDER) are obtained. Existence and uniqueness of strong solution is ensured under the relaxation on the drift coefficient (instead of the Lipschitz character, a monotonicity condition is supposed).

Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results

In this work we study several questions concerning to abstract fractional Cauchy problems of order

Global attractor and omega-limit sets structure for a phase-field model of thermal alloys

\alpha\in(0,1)

Probabilistic representation of solutions for quasi-linear parabolic PDE via FBSDE with reflecting boundary conditions

. Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.

ATTRACTORS FOR PARAMETRIC DELAY DIFFERENTIAL EQUATIONS AND THEIR CONTINUOUS BEHAVIOR

Abstract In this paper, the existence of weak solutions is established for a phase-field model of thermal alloys supplemented with Dirichlet boundary conditions. After that, the existence of global attractors for the associated multi-valued dynamical systems is proved, and the relationship among these sets is established. Finally, we provide a more detailed description of the asymptotic behaviour of solutions via the omega-limit sets. Namely, we obtain a characterization–through the natural stationary system associated to the model–of the elements belonging to the omega-limit sets under suitable assumptions.