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A simple proof of \(L^q\)-estimates for the steady-state Oseen and Stokes equations in a rotating frame. I: strong solutions. (English) Zbl 1261.35106
Summary: 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 [Tohoku Math. J. (2) 58, No. 1, 129–147 (2006; Zbl 1136.76340)], for the Oseen case, and R. Farwig et al. [Pac. J. Math. 215, No. 2, 297–312 (2004; Zbl 1057.35028)], 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 [Zbl 1260.35111] we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space \(D^{-1,q}_0(\mathbb R^3)\).

MSC:
35Q30 Navier-Stokes equations
35B45 A priori estimates in context of PDEs
76D07 Stokes and related (Oseen, etc.) flows
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