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

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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

Reinhard Farwig ^1, and Yasushi Taniuchi ^2,

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

Abstract
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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 $BC(-\infty,T;L^{3}_w\cap L^{6,2})$.

Keywords: unbounded domains., asymptotically almost periodic solutions, uniqueness, Navier-Stokes equations.

Mathematics Subject Classification: 35Q30, 35Q35, 76D0.

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.

show all references

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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