L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

Abstract Consider the equations of Navier-Stokes in the exterior of a rotating domain. It is shown that, after rewriting the problem on a fixed domain Ω, the solution of the corresponding Stokes equation is governed by a C 0-semigroup on L σ p (Ω), 1 < p < ∞, with generator . Moreover, for and initial data u 0 ∈ L σ p (Ω), we prove the existence of a unique local mild solution to the Navier-Stokes problem.

[1]  T. Hishida.  THE STOKES OPERATOR WITH ROTATION EFFECT IN EXTERIOR DOMAINS , 1999 .

[2]  Y. Giga,et al.  regularizing-decay rate estmates for solutions to the Navier-Stokes initial value problem , 2002 .

[3]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[4]  H. Amann.  On the Strong Solvability of the Navier—Stokes Equations , 2000 .

[5]  Yoshikazu Giga,et al.  Analyticity of the semigroup generated by the Stokes operator inLr spaces , 1981 .

[6]  A. Babin,et al.  Global regularity of 3D rotating Navier-Stokes equations for resonant domains , 2000, Appl. Math. Lett..

[7]  Takashi Kato,et al.  StrongLp-solutions of the Navier-Stokes equation inRm, with applications to weak solutions , 1984 .

[8]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[9]  Toshiaki Hishida,et al.  An Existence Theorem¶for the Navier-Stokes Flow¶in the Exterior of a Rotating Obstacle , 1999 .

[10]  Y. Giga,et al.  Global Existence of Two-Dimensional Navier—Stokes Flow with Nondecaying Initial Velocity , 2001 .

[11]  V. A. Solonnikov,et al.  Estimates for solutions of nonstationary Navier-Stokes equations , 1977 .

[12]  Andrew J. Majda,et al.  Vorticity and the mathematical theory of incompressible fluid flow , 1986 .

[13]  Yoshikazu Giga,et al.  Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system , 1986 .

[14]  Marco Cannone,et al.  Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations , 2005 .

[15]  Yoshikazu Giga,et al.  Solutions in Lr of the Navier-Stokes initial value problem , 1985 .

[16]  Zhimin Chen,et al.  Decay properties of weak solutions to a perturbed Navier-Stokes system in Rn , 1997 .

[17]  Matthias Hieber,et al.  On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order , 2006 .

[18]  W. Borchers,et al.  On the semigroup of the Stokes operator for exterior domains inLq-spaces , 1987 .

[19]  R. Farwig,et al.  Lq-theory of a singular "winding'' integral operator arising from fluid dynamics , 2004 .

[20]  Giovanni P. Galdi,et al.  Strong Solutions to the Navier-Stokes Equations Around a Rotating Obstacle , 2005 .

[21]  Matthias Hieber,et al.  The Navier-Stokes Equations in ℝn with Linearly Growing Initial Data , 2005 .

[22]  Y. Giga,et al.  On the Stokes operator in exterior domains , 1988 .

[23]  H. Okamoto.  Exact solutions of the Navier-Stokes equations via Leray’s scheme , 1997 .

[24]  儀我 美一,et al.  Solutions for semilinear parabolic equations in L[p] and regularity of weak solutions of the Navier-Stokes system , 1985 .

[25]  Alex Mahalov,et al.  3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity , 2001 .

[26]  Giovanni P. Galdi,et al.  Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle , 2003 .

[27]  André Noll,et al.  H∞-calculus for the Stokes operator on Lq-spaces , 2003 .

[28]  A. Rhandi,et al.  The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure , 2002 .