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Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

Thierry Gallay — 2012

Journées Équations aux dérivées partielles

We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as t → ∞ . This result was obtained in...

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay;  Romain Joly — 2009

Annales scientifiques de l'École Normale Supérieure

We consider the damped wave equation α u t t + u t = u x x - 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 - s t ) 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 data are sufficiently close to the profile of a front for large | x | , the solution of the damped wave equation converges uniformly on ℝ to a travelling front as t → + ∞ . The proof of this global stability...

Sur le temps de vie de la turbulence bidimensionnelle

Thierry Gallay;  Luis Miguel Rodrigues — 2008

Annales de la faculté des sciences de Toulouse Mathématiques

On sait que toutes les solutions de l’équation de Navier-Stokes dans R 2 dont le tourbillon est intégrable convergent lorsque t → ∞ vers un écoulement autosimilaire appelé tourbillon d’Oseen. Dans cet article, nous donnons une estimation du temps nécessaire à la solution pour atteindre un voisinage du tourbillon d’Oseen à partir d’une donnée initiale arbitraire, mais bien localisée en espace. Nous obtenons ainsi une borne supérieure sur le temps de vie de la turbulence bidimensionnelle libre, en fonction...

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