$L^q$-approach to weak solutions of the Oseen flow around a rotating body

Tom 81 / 2008

Stanislav Kračmar, Šárka Nečasová, Patrick Penel Banach Center Publications 81 (2008), 259-276 MSC: Primary 76D05; Secondary 35Q30. DOI: 10.4064/bc81-0-17


We consider the time-periodic Oseen flow around a rotating body in $\mathbb R^{3}$. We prove a priori estimates in $L^{q}$-spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term $-(\omega\wedge x)\cdot\nabla u+\omega\wedge u$ in the equation of momentum where $\omega$ denotes the angular velocity. We prove the existence of generalized weak solutions in $L^{q}$-space using Littlewood-Paley decomposition and maximal operators.


  • Stanislav KračmarDepartment of Technical Mathematics
    Karlovo nám. 13
    121 35 Prague 2, Czech Republic
  • Šárka NečasováMathematical Institute of Academy of Sciences
    Žitná 25
    11567 Prague 1, Czech Republic
  • Patrick PenelDepartment of Mathematics
    Université de Sud Toulon-Var
    B.P. 20132
    83957 La Garde Cedex, France

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