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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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C. Corduneanu, "Almost Periodic Functions," Interscience Tracts in Pure and Applied Mathematics, No. 22, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-space, Analysis, 16 (1996), 1-26.  Google Scholar

[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.  Google Scholar

[12]

R. Farwig and Y. Taniuchi, Backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. Kubo, The Stokes and Navier-Stokes Equations in an aperture domain, J. Math. Soc. Japan, 59 (2007), 837-859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Y. Meyer, Wavelets, paraproducts, and Navier-Stokes equations, in "Current Developments in Mathematics, 1996" (Cambridge, MA), Int. Press, Boston, MA, (1997), 105-212.  Google Scholar

[32]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. Taniuchi, On stability of periodic solutions of the Navier-Stokes equations in unbounded domains, Hokkaido Math. J., 28 (1999), 147-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-space, Analysis, 16 (1996), 1-26.  Google Scholar

[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.  Google Scholar

[12]

R. Farwig and Y. Taniuchi, Backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. Kubo, The Stokes and Navier-Stokes Equations in an aperture domain, J. Math. Soc. Japan, 59 (2007), 837-859.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

P. Maremonti, Existence and stability of time-periodic solutions to the Navier-Stokes equations in the whole space, Nonlinearity, 4 (1991), 503-529.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Y. Meyer, Wavelets, paraproducts, and Navier-Stokes equations, in "Current Developments in Mathematics, 1996" (Cambridge, MA), Int. Press, Boston, MA, (1997), 105-212.  Google Scholar

[32]

T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. Taniuchi, On stability of periodic solutions of the Navier-Stokes equations in unbounded domains, Hokkaido Math. J., 28 (1999), 147-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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