Thierry Gallay

Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2

Abstract 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 us to extend those results in a number of ways.

Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation

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 solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called “Oseen’s vortex”. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation has several families of quasi-periodic traveling 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 within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.

A center-stable manifold theorem for differential equations in Banach spaces

We prove a center-stable manifold theorem for a class of differential equations in (infinite-dimensional) Banach spaces.

Scaling Variables and Stability of Hyperbolic Fronts

We consider the damped hyperbolic equation

KP description of unidirectional long waves. The model case

\epsilon u_{tt} + u_t \,=\, u_{xx} + \FF(u) , \quad x \in \mathbf{R} , \quad t \ge 0 , \leqno(1)

Front Solutions for the Ginzburg-Landau Equation

where

Three-Dimensional Stability of Burgers Vortices: the Low Reynolds Number Case.

\epsilon