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

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

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

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

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

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

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

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

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

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

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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

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

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

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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