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

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show all references

References:
[1]

Academic Press, New York e.a., 1975.  Google Scholar

[2]

Prikl. Mat. Meh., 37 (1973), 690-705 (Russian); English translation, J. Appl. Math. Mech., 37 (1973), 651-665.  Google Scholar

[3]

DCDS-B, 10 (2008), 1-18. doi: 10.3934/dcdsb.2008.10.1.  Google Scholar

[4]

J. Math. Fluid Mech., 14 (2012), 117-139. doi: 10.1007/s00021-010-0040-z.  Google Scholar

[5]

IASME Transactions, 6 (2005), 900-904.  Google Scholar

[6]

in "Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics" Chalkida, Greeece, (2006), 117-125. Google Scholar

[7]

WSEAS Transactions on Math., 5 (2006), 252-259.  Google Scholar

[8]

Banach Center Publications, 81 (2008), 119-132. doi: 10.4064/bc81-0-8.  Google Scholar

[9]

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]

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]

Math. Nachr., 269/270 (2004), 86-115. doi: 10.1002/mana.200310167.  Google Scholar

[15]

SIAM J. Math. Anal., 43 (2011), 705-738. doi: 10.1137/100786198.  Google Scholar

[16]

Indiana Univ. Math. J., 53 (2004), 1291-1330. doi: 10.1512/iumj.2004.53.2463.  Google Scholar

[17]

J. Math. Fluid Mech., 7 (2005), 339-367. doi: 10.1007/s00021-004-0132-8.  Google Scholar

[18]

Math. Z., 211 (1992), 409-447. doi: 10.1007/BF02571437.  Google Scholar

[19]

Arch. Rational Mech. Anal., 19 (1965), 363-406.  Google Scholar

[20]

Noordhoff, Leyden, 1977.  Google Scholar

[21]

(corr. 2nd print.), Springer, New York e.a., 1998. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[22]

Springer, New York e.a., 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[23]

Acta Math., 129 (1972), 11-34.  Google Scholar

[24]

Indiana Univ. Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048.  Google Scholar

[25]

SIAM J. Math. Anal., 3 (1972), 506-511.  Google Scholar

[26]

in "Approximation Methods for Navier-Stokes Problems" (ed. R. Rautmann), Lecture Notes in Math., 771, Springer (1979), 287-298.  Google Scholar

[27]

Math. Ann., 310 (1998), 1-45. doi: 10.1007/s002080050134.  Google Scholar

[28]

J. Math. Soc. Japan, 53 (2001), 59-111. doi: 10.2969/jmsj/05310059.  Google Scholar

[29]

J. Math. Soc. Japan, 27 (1975), 294-327.  Google Scholar

[30]

SIAM J. Math. Anal., 12 (1981), 201-228. doi: 10.1137/0512021.  Google Scholar

[31]

Hiroshima Math. J., 12 (1982), 115-140.  Google Scholar

[32]

J. Math. Soc. Japan, 36 (1984), 497-522. doi: 10.2969/jmsj/03630497.  Google Scholar

[33]

American J. Math., 113 (1991), 293-373. doi: 10.2307/2374910.  Google Scholar

[34]

Quarterly Appl. Math., 57 (1999), 117-155.  Google Scholar

[35]

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]

Nonlinear Anal., 37 (1999), 751-789. doi: 10.1016/S0362-546X(98)00070-4.  Google Scholar

[37]

AMS Chelsea Publishing, Providence R.I., 2001.  Google Scholar

[38]

(6th ed.), Springer, Berlin e.a., 1980.  Google Scholar

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