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- Thierry Gallay
- 2001

We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R2 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.

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complexvalued 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 amplitude… (More)

- Thierry Gallay
- 2003

Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any… (More)

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)

- Thierry Gallay
- 2001

We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R3. 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)

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator Hǫ = −∂2 x + x + iǫf(x) on L(R), where f is a real-valued function and ǫ > 0 a small parameter. We define Σ(ǫ) as the infimum of the real part of the spectrum of Hǫ, and Ψ(ǫ) −1 as the supremum of the norm of the… (More)

- Thierry Gallay, Alexander Mielke
- J. Nonlinear Science
- 2003

We study a coarsening model describing the dynamics of interfaces in the onedimensional 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 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)

- Thierry Gallay, C Eugene Wayne
- Philosophical transactions. Series A…
- 2002

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)

We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in L(R) 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)