Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain
Thierry Gallay1
1 Institut Fourier Université de Grenoble 1 100, rue des Maths, B.P. 74 38402 Saint-Martin-d’Hères, France
Journées équations aux dérivées partielles (2012), article no. 3, 17 p.

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 collaboration with Y. Maekawa (Kobe University).

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→∞ vers un tourbillon d’Oseen autosimilaire. Ce résultat a été obtenu en collaboration avec Y. Maekawa (Université de Kobe).

DOI: 10.5802/jedp.86
Classification: 35Q30, 35B35, 76D05, 76D17
Keywords: Navier-Stokes equation, long-time behavior, exterior domain
Mot clés : Equation de Navier-Stokes, comportement asymptotique, domaine extérieur
Thierry Gallay 1

1 Institut Fourier Université de Grenoble 1 100, rue des Maths, B.P. 74 38402 Saint-Martin-d’Hères, France 
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Thierry Gallay. Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain. Journées équations aux dérivées partielles (2012), article  no. 3, 17 p. doi : 10.5802/jedp.86. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.86/

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