A simple proof of $L^{q}$-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions

Abstract:

Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity $\xi \in \mathbb {R}^3$ and a non-zero angular velocity $\omega \in \mathbb {R}^3\setminus \{ 0\}$ that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen ($\xi \neq 0$) or Stokes ($\xi =0$) equations in a rotating frame of reference. We will consider the corresponding steady-state whole-space problem. Our main result in this first part concerns elliptic estimates of the solutions in terms of data in $L^{q}(\mathbb {R}^3)$. Such estimates have been established by R. Farwig in Tohoku Math. J., Vol. 58, 2006, for the Oseen case, and R. Farwig, T. Hishida, and D. Müller in Pacific J. Math., Vol. 215 (2), 2004, for the Stokes case. We introduce a new approach resulting in an elementary proof of these estimates. Moreover, our method yields more details on how the constants in the estimates depend on $\xi$ and $\omega$. In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space $D^{-1,q}_0(\mathbb {R}^3)$.