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The nonlinear Schrödinger equation has several families of quasi-periodic travelling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable… (More)

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasi-periodic travelling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x − ct for some c ∈ R. In this paper we investigate the stability of the small… (More)

We construct nite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R 2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways.

We use the vorticity formulation to study the long-time behaviour of solutions to the Navier-Stokes equation on R(3). We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the… (More)

Lesétudes numériques et expérimentales du mouvement des fluides indiquent que, si la vorticité initiale est bien localisée, l'´ ecoulement tendà s'organiser autour de tourbillons isolés. Dans cet article, nous montrons que lesécoulements dont la vorticité est suffisamment localiséé evoluent effectivement vers un tourbillon. Plus précisément, nous montrons… (More)

We consider the damped hyperbolic equation εu tt + u t = u xx + F (u) , x ∈ R , t ≥ 0 , (1) where ε is a positive, not necessarily small parameter. We assume that F (0) = F (1) = 0 and that F is concave on the interval [0, 1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria… (More)

We give a variational proof of global stability for bistable travelling waves of scalar reaction-diffusion equations on the real line. In particular, we recover some of the classical results by P. Fife and J.B. McLeod (1977) without any use of the maximum principle. The method that is illustrated here in the simplest possible setting has been successfully… (More)

We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in constantly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the… (More)

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L 1 (R 2) for positive times is entirely determined by the trace of the vorticity at t = 0, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa, and Osada, and by Kato, this uniqueness… (More)

- Thierry GALLAY
- 2007

We consider the damped wave equation αu tt +u t = u xx −V ′ (u) on the whole real line, where V is a bistable potential. This equation has travelling front solutions of the form u(x, t) = h(x − st) which describe a moving interface between two different steady states of the system, one of which being the global minimum of V. We show that, if the initial… (More)