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

Paul Deuring

doi:10.3934/dcds.2013.33.2757

Previous Article
Asymptotics of the $s$-perimeter as $s\searrow 0$
DCDS Home
This Issue
Next Article
Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

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
Related Papers

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

References:

[1]	R. A. Adams, "Sobolev Spaces,", Academic Press, (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. Google Scholar
[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. 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. 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. Google Scholar
[6]	P. Deuring, The single-layer potential associated with the time-dependent Oseen system,, in, (2006), 117. Google Scholar
[7]	P. Deuring, On volume potentials related to the time-dependent Oseen system,, WSEAS Transactions on Math., 5 (2006), 252. Google Scholar
[8]	P. Deuring, On boundary driven time-dependent Oseen flows,, Banach Center Publications, 81 (2008), 119. doi: 10.4064/bc81-0-8. Google Scholar
[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. Google Scholar
[10]	P. Deuring, Spatial decay of time-dependent Oseen flows,, SIAM J. Math. Anal., 41 (2009), 886. 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. 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. 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. 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. 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. 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. Google Scholar
[20]	S. Fučik, O. John and A. Kufner, "Function Spaces,", Noordhoff, (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.), (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, (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. 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. 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. Google Scholar
[26]	G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations,, in, 771 (1979), 287. Google Scholar
[27]	T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains,, Math. Ann., 310 (1998), 1. 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. 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. Google Scholar
[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. Google Scholar
[31]	T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115. Google Scholar
[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. 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. 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. 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. Google Scholar
[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. Google Scholar
[37]	R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar
[38]	K. Yoshida, "Functional Analysis,", (6th ed.), (1980). Google Scholar

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces,", Academic Press, (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. Google Scholar
[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. 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. 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. Google Scholar
[6]	P. Deuring, The single-layer potential associated with the time-dependent Oseen system,, in, (2006), 117. Google Scholar
[7]	P. Deuring, On volume potentials related to the time-dependent Oseen system,, WSEAS Transactions on Math., 5 (2006), 252. Google Scholar
[8]	P. Deuring, On boundary driven time-dependent Oseen flows,, Banach Center Publications, 81 (2008), 119. doi: 10.4064/bc81-0-8. Google Scholar
[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. Google Scholar
[10]	P. Deuring, Spatial decay of time-dependent Oseen flows,, SIAM J. Math. Anal., 41 (2009), 886. 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. 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. 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. 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. 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. 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. Google Scholar
[20]	S. Fučik, O. John and A. Kufner, "Function Spaces,", Noordhoff, (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.), (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, (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. 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. 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. Google Scholar
[26]	G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations,, in, 771 (1979), 287. Google Scholar
[27]	T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains,, Math. Ann., 310 (1998), 1. 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. 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. Google Scholar
[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. Google Scholar
[31]	T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115. Google Scholar
[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. 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. 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. 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. Google Scholar
[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. Google Scholar
[37]	R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis,", AMS Chelsea Publishing, (2001). Google Scholar
[38]	K. Yoshida, "Functional Analysis,", (6th ed.), (1980). Google Scholar

[1]	Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143
[2]	Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061
[3]	Šárka Nečasová. Stokes and Oseen flow with Coriolis force in the exterior domain. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 339-351. doi: 10.3934/dcdss.2008.1.339
[4]	Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
[5]	Marcello D'Abbicco, Ruy Coimbra Charão, Cleverson Roberto da Luz. Sharp time decay rates on a hyperbolic plate model under effects of an intermediate damping with a time-dependent coefficient. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2419-2447. doi: 10.3934/dcds.2016.36.2419
[6]	Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009
[7]	Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[8]	Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
[9]	Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011
[10]	Takeshi Fukao, Masahiro Kubo. Time-dependent obstacle problem in thermohydraulics. Conference Publications, 2009, 2009 (Special) : 240-249. doi: 10.3934/proc.2009.2009.240
[11]	Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
[12]	Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
[13]	G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
[14]	Jin Takahashi, Eiji Yanagida. Time-dependent singularities in the heat equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 969-979. doi: 10.3934/cpaa.2015.14.969
[15]	Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323
[16]	Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407
[17]	Mourad Choulli, Yavar Kian. Stability of the determination of a time-dependent coefficient in parabolic equations. Mathematical Control & Related Fields, 2013, 3 (2) : 143-160. doi: 10.3934/mcrf.2013.3.143
[18]	Leonardo J. Colombo, María Emma Eyrea Irazú, Eduardo García-Toraño Andrés. A note on Hybrid Routh reduction for time-dependent Lagrangian systems. Journal of Geometric Mechanics, 2020, 12 (2) : 309-321. doi: 10.3934/jgm.2020014
[19]	Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041
[20]	Feng Zhou, Chunyou Sun, Xin Li. Dynamics for the damped wave equations on time-dependent domains. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1645-1674. doi: 10.3934/dcdsb.2018068

2019 Impact Factor: 1.338

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors