TY - JOUR

T1 - A simple proof of Lq-estimates for the steady-state oseen and stokes equations in a rotating frame. parT I

T2 - Strong solutions

AU - Galdi, Giovanni P.

AU - Kyed, Mads

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2013

Y1 - 2013

N2 - Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity ξ ∈ ℝ3 and a non-zero angular velocity ω ∈ ℝ3 \ {0} that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen (ξ ≠ 0) or Stokes (ξ = 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 Lq(ℝ3). 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 ξ and ω. In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space D-1,q 0 (ℝ3).

AB - Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity ξ ∈ ℝ3 and a non-zero angular velocity ω ∈ ℝ3 \ {0} that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen (ξ ≠ 0) or Stokes (ξ = 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 Lq(ℝ3). 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 ξ and ω. In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space D-1,q 0 (ℝ3).

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U2 - 10.1090/S0002-9939-2012-11638-7

DO - 10.1090/S0002-9939-2012-11638-7

M3 - Article

AN - SCOPUS:84870568013

VL - 141

SP - 573

EP - 583

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -