# American Institute of Mathematical Sciences

October  2013, 6(5): 1215-1224. doi: 10.3934/dcdss.2013.6.1215

## Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains

 1 Fachbereich Mathematik and Center of Smart Interfaces (CSI), Technische Universität Darmstadt, 64283 Darmstadt 2 Department of Mathematical Sciences, Shinshu University, Matsumoto 390-8621, Japan

Received  November 2011 Published  March 2013

We present a uniqueness theorem for backward asymptotically almost periodic solutions to the Navier-Stokes equations in $3$-dimensional unbounded domains. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution in $BC(-\infty,T;L^{3}_w)$ within the class of solutions which have sufficiently small $L^{\infty}( L^{3}_w)$-norm. In this paper, we show that a small backward asymptotically almost periodic solution in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$ is unique within the class of all backward asymptotically almost periodic solutions in $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.
Citation: Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215
##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. [2] W. Borchers and T. Miyakawa, $L^2$ decay for Navier-Stokes flow in halfspaces, Math. Ann., 282 (1988), 139-155. doi: 10.1007/BF01457017. [3] W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227. doi: 10.1007/BF02391905. [4] M. Cannone and F. Planchon, On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations, Rev. Mat. Iberoamericana, 16 (2000), 1-16. doi: 10.4171/RMI/268. [5] C. Corduneanu, "Almost Periodic Functions," Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. [6] F. Crispo and P. Maremonti, Navier-Stokes equations in aperture domains: Global existence with bounded flux and time-periodic solutions, Math. Meth. Appl. Sci., 31 (2008), 249-277. doi: 10.1002/mma.903. [7] R. Farwig and T. Okabe, Periodic solutions of the Navier-Stokes equations with inhomogeneous boundary conditions, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (2010), 249-281. doi: 10.1007/s11565-010-0108-y. [8] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643. doi: 10.2969/jmsj/04640607. [9] R. Farwig and H. Sohr, On the Stokes and Navier-Stokes system for domains with noncompact boundary in $L^q$-spaces, Math. Nachr., 170 (1994), 53-77. doi: 10.1002/mana.19941700106. [10] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-space, Analysis, 16 (1996), 1-26. [11] R. Farwig and Y. Taniuchi, Uniqueness of almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains, J. Evol. Equ., 11 (2011), 485-508. doi: 10.1007/s00028-010-0098-3. [12] R. Farwig and Y. Taniuchi, Backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains, preprint. [13] D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 24 (1977), 685-700. [14] G. Furioli, P.-G. Lemarié-Rieusset and E. Terraneo, Sur l'unicité dans $L^3(\mathbbR^3)$ des solutions "mild" des équations de Navier-Stokes, C. R. Acad. Sci. Paris, Sér. I Math., 325 (1997), 1253-1256. doi: 10.1016/S0764-4442(97)82348-8. [15] G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9. [16] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L^r$ spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. [17] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130. [18] H. Iwashita, $L_q-L_r$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in $L_q$ spaces, Math. Ann., 285 (1989), 265-288. doi: 10.1007/BF01443518. [19] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [20] H. Kozono and M. Nakao, Periodic solutions of the the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411. [21] H. Kozono and T. Ogawa, On stability of Navier-Stokes flows in exterior domains, Arch. Ration. Mech. Anal., 128 (1994), 1-31. doi: 10.1007/BF00380792. [22] H. Kozono and M. Yamazaki, Exterior problem for the stationary Navier-Stokes equations in the Lorentz space, Math. Ann., 310 (1998), 279-305. doi: 10.1007/s002080050149. [23] H. Kozono and M. Yamazaki, Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains, Nonlinear Anal., 38 (1999), Ser. A: Theory and Methods, 959-970. doi: 10.1016/S0362-546X(98)00145-X. [24] T. Kubo, The Stokes and Navier-Stokes Equations in an aperture domain, J. Math. Soc. Japan, 59 (2007), 837-859. [25] T. Kubo, Periodic solutions of the Navier-Stokes equations in a perturbed half-space and an aperture domain, Math. Methods Appl. Sci., 28 (2005), 1341-1357. doi: 10.1002/mma.618. [26] T. Kubo and Y. Shibata, On some properties of solutions to the Stokes equation in the half-space and perturbed half-space, in "Dispersive Nonlinear Problems in Mathematical Physics," Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, (2004), 149-220. [27] P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier-Stokes system in $L^N$, Comm. Partial Differential Equations, 26 (2001), 2211-2226. doi: 10.1081/PDE-100107819. [28] P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. [29] P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space, Ric. Mat., 40 (1991), 81-135. [30] P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci. (New York), 93 (1999), 719-746. doi: 10.1007/BF02366850. [31] Y. Meyer, Wavelets, paraproducts, and Navier-Stokes equations, in "Current Developments in Mathematics, 1996" (Cambridge, MA), Int. Press, Boston, MA, (1997), 105-212. [32] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [33] S. Monniaux, Uniqueness of mild solutions of the Navier-Stokes equation and maximal $L^p$-regularity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 663-668. doi: 10.1016/S0764-4442(99)80231-6. [34] R. Salvi, On the existence of periodic weak solutions on the Navier-Stokes equations in exterior regions with periodically moving boundaries, in "Navier-Stokes Equations and Related Nonlinear Problems" (ed. A. Sequeira) (Funchal, 1994), Plenum, New York, (1995), 63-73. [35] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation" (ed. G. P. Galdi), Series Adv. Math. Appl. Sci., 11, World Scientific, River Edge, NJ, (1992), 1-35. [36] Y. Taniuchi, On stability of periodic solutions of the Navier-Stokes equations in unbounded domains, Hokkaido Math. J., 28 (1999), 147-173. [37] Y. Taniuchi, On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains, Math. Z., 261 (2009), 597-615. doi: 10.1007/s00209-008-0341-6. [38] S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$, Comm. Pure Appl. Math., 40 (1987), 611-621. doi: 10.1002/cpa.3160400506. [39] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.

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##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, "Vector-Valued Laplace Transforms and Cauchy Problems," Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. [2] W. Borchers and T. Miyakawa, $L^2$ decay for Navier-Stokes flow in halfspaces, Math. Ann., 282 (1988), 139-155. doi: 10.1007/BF01457017. [3] W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227. doi: 10.1007/BF02391905. [4] M. Cannone and F. Planchon, On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations, Rev. Mat. Iberoamericana, 16 (2000), 1-16. doi: 10.4171/RMI/268. [5] C. Corduneanu, "Almost Periodic Functions," Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. [6] F. Crispo and P. Maremonti, Navier-Stokes equations in aperture domains: Global existence with bounded flux and time-periodic solutions, Math. Meth. Appl. Sci., 31 (2008), 249-277. doi: 10.1002/mma.903. [7] R. Farwig and T. Okabe, Periodic solutions of the Navier-Stokes equations with inhomogeneous boundary conditions, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (2010), 249-281. doi: 10.1007/s11565-010-0108-y. [8] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643. doi: 10.2969/jmsj/04640607. [9] R. Farwig and H. Sohr, On the Stokes and Navier-Stokes system for domains with noncompact boundary in $L^q$-spaces, Math. Nachr., 170 (1994), 53-77. doi: 10.1002/mana.19941700106. [10] R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-space, Analysis, 16 (1996), 1-26. [11] R. Farwig and Y. Taniuchi, Uniqueness of almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains, J. Evol. Equ., 11 (2011), 485-508. doi: 10.1007/s00028-010-0098-3. [12] R. Farwig and Y. Taniuchi, Backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains, preprint. [13] D. Fujiwara and H. Morimoto, An $L_r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 24 (1977), 685-700. [14] G. Furioli, P.-G. Lemarié-Rieusset and E. Terraneo, Sur l'unicité dans $L^3(\mathbbR^3)$ des solutions "mild" des équations de Navier-Stokes, C. R. Acad. Sci. Paris, Sér. I Math., 325 (1997), 1253-1256. doi: 10.1016/S0764-4442(97)82348-8. [15] G. P. Galdi and H. Sohr, Existence and uniqueness of time-periodic physically reasonable Navier-Stokes flow past a body, Arch. Ration. Mech. Anal., 172 (2004), 363-406. doi: 10.1007/s00205-004-0306-9. [16] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $L^r$ spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. [17] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130. [18] H. Iwashita, $L_q-L_r$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in $L_q$ spaces, Math. Ann., 285 (1989), 265-288. doi: 10.1007/BF01443518. [19] T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182. [20] H. Kozono and M. Nakao, Periodic solutions of the the Navier-Stokes equations in unbounded domains, Tohoku Math. J. (2), 48 (1996), 33-50. doi: 10.2748/tmj/1178225411. [21] H. Kozono and T. Ogawa, On stability of Navier-Stokes flows in exterior domains, Arch. Ration. Mech. Anal., 128 (1994), 1-31. doi: 10.1007/BF00380792. [22] H. Kozono and M. Yamazaki, Exterior problem for the stationary Navier-Stokes equations in the Lorentz space, Math. Ann., 310 (1998), 279-305. doi: 10.1007/s002080050149. [23] H. Kozono and M. Yamazaki, Uniqueness criterion of weak solutions to the stationary Navier-Stokes equations in exterior domains, Nonlinear Anal., 38 (1999), Ser. A: Theory and Methods, 959-970. doi: 10.1016/S0362-546X(98)00145-X. [24] T. Kubo, The Stokes and Navier-Stokes Equations in an aperture domain, J. Math. Soc. Japan, 59 (2007), 837-859. [25] T. Kubo, Periodic solutions of the Navier-Stokes equations in a perturbed half-space and an aperture domain, Math. Methods Appl. Sci., 28 (2005), 1341-1357. doi: 10.1002/mma.618. [26] T. Kubo and Y. Shibata, On some properties of solutions to the Stokes equation in the half-space and perturbed half-space, in "Dispersive Nonlinear Problems in Mathematical Physics," Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta, (2004), 149-220. [27] P.-L. Lions and N. Masmoudi, Uniqueness of mild solutions of the Navier-Stokes system in $L^N$, Comm. Partial Differential Equations, 26 (2001), 2211-2226. doi: 10.1081/PDE-100107819. [28] P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529. [29] P. Maremonti, Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary conditions in half-space, Ric. Mat., 40 (1991), 81-135. [30] P. Maremonti and M. Padula, Existence, uniqueness, and attainability of periodic solutions of the Navier-Stokes equations in exterior domains, J. Math. Sci. (New York), 93 (1999), 719-746. doi: 10.1007/BF02366850. [31] Y. Meyer, Wavelets, paraproducts, and Navier-Stokes equations, in "Current Developments in Mathematics, 1996" (Cambridge, MA), Int. Press, Boston, MA, (1997), 105-212. [32] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [33] S. Monniaux, Uniqueness of mild solutions of the Navier-Stokes equation and maximal $L^p$-regularity, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 663-668. doi: 10.1016/S0764-4442(99)80231-6. [34] R. Salvi, On the existence of periodic weak solutions on the Navier-Stokes equations in exterior regions with periodically moving boundaries, in "Navier-Stokes Equations and Related Nonlinear Problems" (ed. A. Sequeira) (Funchal, 1994), Plenum, New York, (1995), 63-73. [35] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation" (ed. G. P. Galdi), Series Adv. Math. Appl. Sci., 11, World Scientific, River Edge, NJ, (1992), 1-35. [36] Y. Taniuchi, On stability of periodic solutions of the Navier-Stokes equations in unbounded domains, Hokkaido Math. J., 28 (1999), 147-173. [37] Y. Taniuchi, On the uniqueness of time-periodic solutions to the Navier-Stokes equations in unbounded domains, Math. Z., 261 (2009), 597-615. doi: 10.1007/s00209-008-0341-6. [38] S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$, Comm. Pure Appl. Math., 40 (1987), 611-621. doi: 10.1002/cpa.3160400506. [39] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.
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