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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

Paul Deuring 1,

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.
Keywords: time-dependent Oseen system, pointwise decay., Exterior domain.
Mathematics Subject Classification: Primary: 35Q30, 65N30; Secondary: 76D0.
Citation: Paul Deuring. Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259.  Google Scholar

[8]

P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132. doi: 10.4064/bc81-0-8.  Google Scholar

[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.  Google Scholar

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P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922. doi: 10.1137/080723831.  Google Scholar

[11]

P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., ().   Google Scholar

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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., ().   Google Scholar

[13]

P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406.  Google Scholar

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S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34.  Google Scholar

[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.  Google Scholar

[25]

G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $ \mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001.  Google Scholar

[38]

K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980.  Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, New York e.a., 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[7]

P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259.  Google Scholar

[8]

P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132. doi: 10.4064/bc81-0-8.  Google Scholar

[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.  Google Scholar

[10]

P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922. doi: 10.1137/080723831.  Google Scholar

[11]

P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., ().   Google Scholar

[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., ().   Google Scholar

[13]

P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34.  Google Scholar

[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.  Google Scholar

[25]

G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $ \mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001.  Google Scholar

[38]

K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980.  Google Scholar

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