The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: The\n non-autonomous case

2010/06/18 by Tobias Hansel, Hansel, Tobias, Abdelaziz Rhandi +1
Mathematics · Engineering · #Navier-Stokes equation solutions #Advanced Mathematical Physics Problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1006.3643

Abstract

Consider the Navier-Stokes flow past a rotating obstacle with a general\ntime-dependent angular velocity and a time-dependent outflow condition at\ninfinity -- sometimes called an Oseen condition. By a suitable change of\ncoordinates the problem is transformed to an non-autonomous problem with\nunbounded drift terms on a fixed exterior domain \Ω\⊂ R^d. It is\nshown that the solution to the linearized problem is governed by a strongly\ncontinuous evolution system T_\Ω(t,s) _{t\≥ s\≥0} on\nL^p_\σ(\Ω) for 1<p<\∞. Moreover, L^p-L^q smoothing\nproperties and gradient estimates of T_\Ω(t,s), 0\≤ s \≤ t, are\nobtained. These results are the key ingredients to show local in time existence\nof mild solutions to the full nonlinear problem for p\≥ d and initial value\nin L^p_\σ(\Ω).\n

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