# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2013, 33 (7) : 2757-2776. doi: 10.3934/dcds.2013.33.2757
##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (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. [3] H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations,, DCDS-B, 10 (2008), 1. 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. 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. [6] P. Deuring, The single-layer potential associated with the time-dependent Oseen system,, in, (2006), 117. [7] P. Deuring, On volume potentials related to the time-dependent Oseen system,, WSEAS Transactions on Math., 5 (2006), 252. [8] P. Deuring, On boundary driven time-dependent Oseen flows,, Banach Center Publications, 81 (2008), 119. doi: 10.4064/bc81-0-8. [9] P. Deuring, A potential theoretic approach to the time-dependent Oseen system,, in, (2010), 191. 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. 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. 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. 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. 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. 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. 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. [20] S. Fučik, O. John and A. Kufner, "Function Spaces,", Noordhoff, (1977). [21] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems,", (corr. 2nd print.), (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, (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. [24] J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions,, Indiana Univ. Math. J., 29 (1980), 639. 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. [26] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations,, in, 771 (1979), 287. [27] T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains,, Math. Ann., 310 (1998), 1. 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. 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. [30] M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$,, SIAM J. Math. Anal., 12 (1981), 201. 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. [32] R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies,, J. Math. Soc. Japan, 36 (1984), 497. 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. doi: 10.2307/2374910. [34] Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations,, Quarterly Appl. Math., 57 (1999), 117. [35] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153. [36] S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations,, Nonlinear Anal., 37 (1999), 751. doi: 10.1016/S0362-546X(98)00070-4. [37] R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). [38] K. Yoshida, "Functional Analysis,", (6th ed.), (1980).

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##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (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. [3] H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations,, DCDS-B, 10 (2008), 1. 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. 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. [6] P. Deuring, The single-layer potential associated with the time-dependent Oseen system,, in, (2006), 117. [7] P. Deuring, On volume potentials related to the time-dependent Oseen system,, WSEAS Transactions on Math., 5 (2006), 252. [8] P. Deuring, On boundary driven time-dependent Oseen flows,, Banach Center Publications, 81 (2008), 119. doi: 10.4064/bc81-0-8. [9] P. Deuring, A potential theoretic approach to the time-dependent Oseen system,, in, (2010), 191. 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. 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. 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. 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. 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. 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. 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. [20] S. Fučik, O. John and A. Kufner, "Function Spaces,", Noordhoff, (1977). [21] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems,", (corr. 2nd print.), (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, (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. [24] J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions,, Indiana Univ. Math. J., 29 (1980), 639. 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. [26] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations,, in, 771 (1979), 287. [27] T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains,, Math. Ann., 310 (1998), 1. 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. 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. [30] M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$,, SIAM J. Math. Anal., 12 (1981), 201. 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. [32] R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies,, J. Math. Soc. Japan, 36 (1984), 497. 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. doi: 10.2307/2374910. [34] Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations,, Quarterly Appl. Math., 57 (1999), 117. [35] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153. [36] S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations,, Nonlinear Anal., 37 (1999), 751. doi: 10.1016/S0362-546X(98)00070-4. [37] R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). [38] K. Yoshida, "Functional Analysis,", (6th ed.), (1980).
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