MR3114825 Tomoyuki NAKATSUKA Tomoyuki NAKATSUKA Uniqueness of Steady Navier-Stokes Flows in Exterior Domains Funkcialaj Ekvacioj. Serio Internacia 56 2013 323--337 http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-2/56_323.pdf http://www.ams.org/mathscinet-getitem?mr=MR3114825 We consider the uniqueness of stationary solutions to the Navier-Stokes equation in $3$-dimensional exterior domains within the class $u\in L_{3,\infty}$ with $\nabla u \in L_{3/2,\infty}$, where $L_{3,\infty}$ and $L_{3/2,\infty}$ are the Lorentz spaces. It is shown that if solutions $u$ and $v$ satisfy the conditions that $u$ is small in $L_{3,\infty}$ and $v \in L_3 + L_{\infty}$, then $u=v$. The proof relies upon the regularity theory for the perturbed Stokes equation. Navier-Stokes flow, Exterior domain, Uniqueness, Lorentz space. 35Q30, 76D05. 56-323 2013 Uniqueness of Steady Navier-Stokes Flows in Exterior Domains Tomoyuki NAKATSUKA Tomoyuki NAKATSUKA 1 Adams, R. A.; Fournier, J. J. F. Sobolev Spaces Second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam 2003 MR2424078 2 Bergh, J.; Löfström, J. Interpolation Spaces Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York 1976 MR0482275
3 Borchers, W.; Miyakawa, T. Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains Acta Math. 165 1990 189-227 MR1075041
4 Farwig, R.; Taniuchi, Y. Uniqueness of almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains J. Evol. Equ. 11 2011 485-508 MR2827097
5 Galdi, G. P. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II: Nonlinear Steady Problems Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York 1994 MR1284206
6 Galdi, G. P.; Simader, C. G. Existence, uniqueness and $L^q$-estimates for the Stokes problem in an exterior domains Arch. Rational Mech. Anal. 112 1990 291-318 MR1077262
7 Galdi, G. P.; Sohr, H. Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body Arch. Ration. Mech. Anal. 172 2004 363-406 MR2062429
8 Kozono, H.; Sohr, H. On a new class of generalized solutions for the Stokes equations in exterior domains Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 1992 155-181 MR1197210
9 Kozono, H.; Yamazaki, M. Exterior problem for the stationary Navier-Stokes equations in the Lorentz space Math. Ann. 310 1998 279-305 MR1602012
10 Kozono, H.; Yamazaki, M. Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains Nonlinear Anal. 38 1999 959-970 MR1716426
11 Miyakawa, T. On uniqueness of steady Navier-Stokes flows in an exterior domain Adv. Math. Sci. Appl. 5 1995 411-420 MR1360998
12 Nakatsuka, T. On uniqueness of stationary solutions to the Navier-Stokes equations in exterior domains Nonlinear Anal. 75 2012 3457-3464 MR2901328
13 Shibata, Y.; Yamazaki, M. Uniform estimates in the velocity at infinity for stationary solutions to the Navier-Stokes exterior problem Japan. J. Math. (N.S.) 31 2005 225-279 MR2198178
14 Taniuchi, Y. On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains Math. Z. 261 2009 597-615 MR2471090