Geneviève Raugel

Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation

Abstract For a smooth bounded domain Ω ⊂R n , n ⩽ 3, and e a real parameter, consider the hyperbolic equation eu tt + u t − Δu = −ƒ(u) − g in Ω with Dirichlet boundary conditions. Under certain conditions on ƒ, this equation has a compact attractor A e in H 0 1 × L 2 . For e = 0, the parabolic equation also has a compact attractor which can be naturally embedded into a compact set A 0 in H 0 1 × L 2 . It is shown that, for any neighborhood U of A 0 , the set A e ⊂ U for e small.

Upper semicontinuity of attractors for approximations of semigroups and partial differential equations

Abstract : Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations.

Convergence in gradient-like systems with applications to PDE

In gradient-like systems, the limit set of an orbit belongs to the set of equilibrium points. We give easily applied conditions to determine when this limit set is a single point. Applications are given to parabolic equations, linearly damped hyperbolic equations as well as their discretizations.

Lower semicontinuity of attractors of gradient systems and applications

SummaryFor 0⩽ε⩽ε0, let Tε(t), t⩾0, be a family of semigroups on a Banach space X with local attractors Aε. Under the assumptions that T0(t) is a gradient system with hyperbolic equilibria and Tε(t) converges to T0(t) in an appropriate sense, it is shown that the attractors {Aε, 0⩽ε⩽ε0} are lower-semicontinuous at zero. Applications are given to ordinary and functional differential equations, parabolic partial differential equations and their space and time discretizations. We also give an estimate of the Hausdorff distance between Aε and A0, in some examples.

A damped hyperbolic equation on thin domains

For a damped hyperbolic equation in a thin domain in R3 over a bounded smooth domain in R2 , it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.

Stability of travelling waves for a damped hyperbolic equation

Abstract. We consider a nonlinear damped hyperbolic equation in

Regularity, determining modes and Galerkin methods

{\bf R}^n, 1 \le n \le 4

A Conforming Finite Element Method for the Time-Dependent Navier–Stokes Equations

, depending on a positive parameter

Partial Differential Equations on Thin Domains

\epsilon

Methodes d'elements finis mixtes pour les equations de stokes et de Navier-Stokes dans un polygone non convexe

. If