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

Gallay, Thierry

Journées équations aux dérivées partielles (2012), article no. 3, 17 p.

Nous étudions le comportement asymptotique en temps des solutions de l’équation de Navier-Stokes incompressible dans un domaine extérieur du plan, avec condition de non-glissement à la frontière. Les données initiales que nous considérons sont des perturbations d’énergie finie d’un tourbillon régulier dont la circulation à l’infini est petite, mais nous n’imposons aucune autre restriction à leur taille. En utilisant une estimation d’énergie logarithmique et des arguments d’interpolation, nous montrons que la solution converge lorsque $t\to \infty$ vers un tourbillon d’Oseen autosimilaire. Ce résultat a été obtenu en collaboration avec Y. Maekawa (Université de Kobe).

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 $t\to \infty$. This result was obtained in collaboration with Y. Maekawa (Kobe University).

Classification : 35Q30, 35B35, 76D05, 76D17
Keywords: Navier-Stokes equation, long-time behavior, exterior domain
Mot clés : Equation de Navier-Stokes, comportement asymptotique, domaine extérieur

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