Thierry Gallay

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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)
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 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 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)
Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator H ǫ = −∂ 2 x + x 2 + iǫ −1 f (x) on L 2 (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 H ǫ , and Ψ(ǫ) −1 as the supremum of the(More)
We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter ν, and we choose a time T > 0 such that the(More)
Burgers vortices are explicit stationary solutions of the Navier-Stokes equations which are often used to describe the vortex tubes observed in numerical simulations of three-dimensional turbulence. In this model, the velocity field is a two-dimensional perturbation of a linear straining flow with axial symmetry. The only free parameter is the Reynolds(More)
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