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 . 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 vers un tourbillon d’Oseen autosimilaire. Ce résultat a été obtenu en collaboration avec Y. Maekawa (Université de Kobe).
Keywords: Navier-Stokes equation, long-time behavior, exterior domain
Mot clés : Equation de Navier-Stokes, comportement asymptotique, domaine extérieur
@article{JEDP_2012____A3_0, author = {Gallay, Thierry}, title = {Long-Time {Asymptotics} for the {Navier-Stokes} {Equation} in a {Two-Dimensional} {Exterior} {Domain}}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {3}, pages = {1--17}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2012}, doi = {10.5802/jedp.86}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.86/} }
TY - JOUR AU - Gallay, Thierry TI - Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain JO - Journées équations aux dérivées partielles PY - 2012 SP - 1 EP - 17 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.86/ DO - 10.5802/jedp.86 LA - en ID - JEDP_2012____A3_0 ER -
%0 Journal Article %A Gallay, Thierry %T Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain %J Journées équations aux dérivées partielles %D 2012 %P 1-17 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.86/ %R 10.5802/jedp.86 %G en %F JEDP_2012____A3_0
Gallay, Thierry. 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. http://www.numdam.org/articles/10.5802/jedp.86/
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