# American Institute of Mathematical Sciences

July  2013, 33(7): 2757-2776. doi: 10.3934/dcds.2013.33.2757

## Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support

 1 Univ Lille Nord de France, 59000 Lille

Received  April 2012 Revised  November 2012 Published  January 2013

The article deals with the time-dependent Oseen system in a 3D exterior domain. It is shown that the velocity part of a weak solution to that system decays as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-1}$, and its spatial gradient as $\bigl(\, |x| \cdot (1+|x|-x_1) \,\bigr) ^{-3/2}$, for $|x|\to \infty$. This result is obtained for data that need not have compact support.
Citation: Paul Deuring. Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2757-2776. doi: 10.3934/dcds.2013.33.2757
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Academic Press, New York e.a., 1975. [2] K. I. Babenko and M. M. Vasil'ev, On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body, Prikl. Mat. Meh., 37 (1973), 690-705 (Russian); English translation, J. Appl. Math. Mech., 37 (1973), 651-665. [3] H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations, DCDS-B, 10 (2008), 1-18. doi: 10.3934/dcdsb.2008.10.1. [4] H.-O. Bae and J. Roh, Stability for the 3D Navier-Stokes equations with nonzero far field velocity on exterior domains, J. Math. Fluid Mech., 14 (2012), 117-139. doi: 10.1007/s00021-010-0040-z. [5] P. Deuring, Exterior stationary Navier-Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient, IASME Transactions, 6 (2005), 900-904. [6] P. Deuring, The single-layer potential associated with the time-dependent Oseen system, in "Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics" Chalkida, Greeece, (2006), 117-125. [7] P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259. [8] P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132. doi: 10.4064/bc81-0-8. [9] P. Deuring, A potential theoretic approach to the time-dependent Oseen system, in "Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday" (eds. R. Rannacher and A. Sequeira), Springer (2010), 191-214. doi: 10.1007/978-3-642-04068-9_12. [10] P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922. doi: 10.1137/080723831. [11] P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., (). [12] P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $\mathbbR^3$: Spatial decay of the velocity,, to appear in Mathematica Bohemica., (). [13] P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., (). [14] P. Deuring and S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains, Math. Nachr., 269/270 (2004), 86-115. doi: 10.1002/mana.200310167. [15] P. Deuring, S. Kračmar and Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and tranlating bodies, SIAM J. Math. Anal., 43 (2011), 705-738. doi: 10.1137/100786198. [16] Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domain, Indiana Univ. Math. J., 53 (2004), 1291-1330. doi: 10.1512/iumj.2004.53.2463. [17] Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation, J. Math. Fluid Mech., 7 (2005), 339-367. doi: 10.1007/s00021-004-0132-8. [18] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447. doi: 10.1007/BF02571437. [19] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. [20] S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977. [21] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems," (corr. 2nd print.), Springer, New York e.a., 1998. doi: 10.1007/978-1-4612-5364-8. [22] G. P. Galdi, "An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer, New York e.a., 1994. doi: 10.1007/978-1-4612-5364-8. [23] J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34. [24] J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048. [25] G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $\mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511. [26] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Problems" (ed. R. Rautmann), Lecture Notes in Math., 771, Springer (1979), 287-298. [27] T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains, Math. Ann., 310 (1998), 1-45. doi: 10.1007/s002080050134. [28] S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $L^p$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111. doi: 10.2969/jmsj/05310059. [29] K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327. [30] M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$, SIAM J. Math. Anal., 12 (1981), 201-228. doi: 10.1137/0512021. [31] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [32] R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 36 (1984), 497-522. doi: 10.2969/jmsj/03630497. [33] Zongwei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, American J. Math., 113 (1991), 293-373. doi: 10.2307/2374910. [34] Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155. [35] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231 (Russian); English translation, J. Soviet Math., 8 (1977), 467-529. [36] S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789. doi: 10.1016/S0362-546X(98)00070-4. [37] R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001. [38] K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Academic Press, New York e.a., 1975. [2] K. I. Babenko and M. M. Vasil'ev, On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body, Prikl. Mat. Meh., 37 (1973), 690-705 (Russian); English translation, J. Appl. Math. Mech., 37 (1973), 651-665. [3] H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations, DCDS-B, 10 (2008), 1-18. doi: 10.3934/dcdsb.2008.10.1. [4] H.-O. Bae and J. Roh, Stability for the 3D Navier-Stokes equations with nonzero far field velocity on exterior domains, J. Math. Fluid Mech., 14 (2012), 117-139. doi: 10.1007/s00021-010-0040-z. [5] P. Deuring, Exterior stationary Navier-Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient, IASME Transactions, 6 (2005), 900-904. [6] P. Deuring, The single-layer potential associated with the time-dependent Oseen system, in "Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics" Chalkida, Greeece, (2006), 117-125. [7] P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259. [8] P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132. doi: 10.4064/bc81-0-8. [9] P. Deuring, A potential theoretic approach to the time-dependent Oseen system, in "Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday" (eds. R. Rannacher and A. Sequeira), Springer (2010), 191-214. doi: 10.1007/978-3-642-04068-9_12. [10] P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922. doi: 10.1137/080723831. [11] P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., (). [12] P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $\mathbbR^3$: Spatial decay of the velocity,, to appear in Mathematica Bohemica., (). [13] P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., (). [14] P. Deuring and S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains, Math. Nachr., 269/270 (2004), 86-115. doi: 10.1002/mana.200310167. [15] P. Deuring, S. Kračmar and Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and tranlating bodies, SIAM J. Math. Anal., 43 (2011), 705-738. doi: 10.1137/100786198. [16] Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domain, Indiana Univ. Math. J., 53 (2004), 1291-1330. doi: 10.1512/iumj.2004.53.2463. [17] Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation, J. Math. Fluid Mech., 7 (2005), 339-367. doi: 10.1007/s00021-004-0132-8. [18] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447. doi: 10.1007/BF02571437. [19] R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. [20] S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977. [21] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems," (corr. 2nd print.), Springer, New York e.a., 1998. doi: 10.1007/978-1-4612-5364-8. [22] G. P. Galdi, "An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer, New York e.a., 1994. doi: 10.1007/978-1-4612-5364-8. [23] J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34. [24] J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048. [25] G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $\mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511. [26] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Problems" (ed. R. Rautmann), Lecture Notes in Math., 771, Springer (1979), 287-298. [27] T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains, Math. Ann., 310 (1998), 1-45. doi: 10.1007/s002080050134. [28] S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $L^p$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111. doi: 10.2969/jmsj/05310059. [29] K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327. [30] M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$, SIAM J. Math. Anal., 12 (1981), 201-228. doi: 10.1137/0512021. [31] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. [32] R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 36 (1984), 497-522. doi: 10.2969/jmsj/03630497. [33] Zongwei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, American J. Math., 113 (1991), 293-373. doi: 10.2307/2374910. [34] Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155. [35] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231 (Russian); English translation, J. Soviet Math., 8 (1977), 467-529. [36] S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789. doi: 10.1016/S0362-546X(98)00070-4. [37] R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001. [38] K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980.
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