Asymptotic profiles for the third grade fluids equations |
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| Affiliation: | 1. LMAP UMR CNRS 5142, IPRA BP 1155, 64013 Pau Cedex, France;2. Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany;1. School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong 519082, PR China;2. Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5 Canada;3. Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada;1. University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie WY 82071, United States;2. SAMM, EA 4543, Université Paris 1 Panthéon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex, France;3. Laboratoire de Probabilités et Modèles Aléatoires, Universités Paris 6–Paris 7, France |
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| Abstract: | We study the long time behaviour of the solutions of the third grade fluids equations in dimension 2. Introducing scaled variables and performing several energy estimates in weighted Sobolev spaces, we describe the first order of an asymptotic expansion of these solutions. It shows in particular that, under smallness assumptions on the data, the solutions of the third grade fluids equations converge to self-similar solutions of the heat equations, which can be computed explicitly from the data. |
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| Keywords: | Fluid mechanics Third grade fluids Asymptotic expansion |
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