Jean-Michel Ghidaglia

Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time

Abstract It is shown that a weak dissipation (of order zero) is sufficient to impose a finite dimensional behavior for Korteweg-de Vries equations with periodic boundary conditions. A universal attractor which captures all the trajectories is constructed. Its fractal dimension is proved to be finite.

Attractors for the penalized Navier-Stokes equations

We consider the penalized form of the Navier–Stokes equations for a viscous incompressible fluid where the pressure and the incompressibility equation

LOWER BOUND ON THE DIMENSION OF THE ATTRACTOR FOR THE NAVIER-STOKES EQUATIONS IN SPACE DIMENSION 3

{\operatorname{div}}\,u = 0

LONG TIME BEHAVIOR FOR PARTLY DISSIPATIVE EQUATIONS: THE SLIGHTLY COMPRESSIBLE 2D-NAVIER-STOKES EQUATIONS

are suppressed and replaced by a penalty term in the momentum conservation equation. In this article we study the existence of an attractor for the penalized Navier–Stokes equation, this attractor describing the long-time behaviour of the solutions. Then we let the penalty parameter tend to zero and we show how the attractors of the penalized equations approximate the attractor of the exact equations.

Long time behaviour of solutions of abstract inequalities: applications to thermo-hydraulic and magnetohydrodynamic equations

Publisher Summary In recent years, there has been a lot of attention on the study of the attractor for the Navier–Stokes equations. This chapter focuses on the results concerning the derivation of a lower bound on the dimension of the attractor for the Navier-–Stokes equations in space dimension 3. The results on the attractor for the two-dimensional equations are also reviewed in the chapter. It presents a comparison of lower bounds and upper bounds and reviews some technical inequalities related to the eigenvalues of the Laplacian and the Sobolev inequality. Dimension 2 and some results concerning the Orr–Sommerfeld equation depending on a parameter are also analyzed in the chapter.

Dimension of the universal attractor describing the periodically driven Sine-Gordon equations

It is known that for dissipative evolution equations, the long time behavior of the solutions is generally described by a compact attractor to which all solutions converge, while such a result is not true for conservative equations of Hamiltonian type. In this paper we consider a partly dissipative system corresponding to the equations of slightly compressible fluids and investigate the long time behavior of their solutions. Despite the lack of compactness and smoothing effect for the pressure variable, the existence of a global attractor is shown and its fractal dimension is estimated.

A note on the strong convergence towards attractors of damped forced KdV equations

Abstract We study some scalar inequalities of parabolic type and we give the leading term of an asymptotic expansion as t → ∞ for solutions of thermo-hydraulic equations without external excitation. A phenomenon of resonance is pointed out. We also treat M. H. D. equations and Navier-Stokes equations on a Riemannian manifold.

Generalization of the Sobolev-Lieb-Thirring inequalities and applications to the dimension of attractors

Abstract In this work we give a rigorous and explicit bound of the fractal dimension of the universal attractor describing the long time behavior of the damped Sine-Gordon equation driven by a time-periodic force. The dependence of the dimension in terms of the physical quantities is provided.