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

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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

Abstract
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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 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.

show all references

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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