Long-time asymptotics of the second grade fluid equations \(\mathbb R^2\) (Q448901)
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scientific article; zbMATH DE number 6079140
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Long-time asymptotics of the second grade fluid equations \(\mathbb R^2\) |
scientific article; zbMATH DE number 6079140 |
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Long-time asymptotics of the second grade fluid equations \(\mathbb R^2\) (English)
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7 September 2012
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second grade fluid system
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weighted Sobolev spaces
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Oseen vortex
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The author considers the system of partial differential equations describing the flow of second grade fluid in the full two-dimensional space. For sufficiently small data, he constructs regular global in time solutions to the given system and shows that the solutions converge to the Oseen vortex NEWLINE\[NEWLINE G(\xi) = C e^{-|\xi|^2/4}. NEWLINE\]NEWLINENEWLINENEWLINEThe solutions are construced by means of regularization of the equation for the vorticity (which is a scalar quantity in two space dimensions). For this approximation, he shows the decay to the Oseen vortex, uniformly with respect to the approximation parameter. The main tool for this are the weighted Sobolev spaces. Finally, the limit passage allows to show the same result for the original problem, the velocity is then reconstructed by means of the Biot-Savart law.
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