\(L^q\)-estimates for the stationary Oseen equations on the exterior of a rotating obstacle. (English) Zbl 1398.35175
Summary: We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate \(L^q\)-estimates. The uniqueness of very weak solutions is shown by the method of cut-off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete \(D^{1,r}\)-result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where \(4/3<r<4\). Here, \(D^{1,r}\) is the homogeneous Sobolev space.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76U05 | General theory of rotating fluids |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
\(L^q\)-estimates; rotating obstacles; stationary Oseen equations; very weak solutions; weak solutionsReferences:
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