The Cauchy problem for the homogeneous time-dependent Oseen system in ℝ 3 : spatial decay of the velocity

Paul Deuring

Mathematica Bohemica (2013)

  • Volume: 138, Issue: 3, page 299-324
  • ISSN: 0862-7959

Abstract

top
We consider the homogeneous time-dependent Oseen system in the whole space ℝ 3 . The initial data is assumed to behave as O ( | x | - 1 - ϵ ) , and its gradient as O ( | x | - 3 / 2 - ϵ ) , when | x | tends to infinity, where ϵ is a fixed positive number. Then we show that the velocity u decays according to the equation | u ( x , t ) | = O ( | x | - 1 ) , and its spatial gradient ∇ x u decreases with the rate | x | - 3 / 2 , for | x | tending to infinity, uniformly with respect to the time variable t . Since these decay rates are optimal even in the stationary case, they should also be the best possible in the evolutionary case considered in this article. We also treat the case ϵ = 0 . Then the preceding decay rates of u remain valid, but they are no longer uniform with respect to t .

How to cite

top

Deuring, Paul. "The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbb {R}^3 $: spatial decay of the velocity." Mathematica Bohemica 138.3 (2013): 299-324. <http://eudml.org/doc/260628>.

@article{Deuring2013,
abstract = {We consider the homogeneous time-dependent Oseen system in the whole space $ \mathbb \{R\}^3 $. The initial data is assumed to behave as $O(|x|^\{-1- \epsilon \})$, and its gradient as $O(|x|^\{-3/2- \epsilon \})$, when $|x|$ tends to infinity, where $\epsilon $ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $|u(x,t)|=O(|x|^\{-1\})$, and its spatial gradient $\nabla _xu$ decreases with the rate $|x|^\{-3/2\}$, for $|x|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible in the evolutionary case considered in this article. We also treat the case $\epsilon =0$. Then the preceding decay rates of $u$ remain valid, but they are no longer uniform with respect to $t$.},
author = {Deuring, Paul},
journal = {Mathematica Bohemica},
keywords = {Cauchy problem; time-dependent Oseen system; spatial decay; wake; Cauchy problem; time-dependent Oseen system; spatial decay; wake},
language = {eng},
number = {3},
pages = {299-324},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbb \{R\}^3 $: spatial decay of the velocity},
url = {http://eudml.org/doc/260628},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Deuring, Paul
TI - The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbb {R}^3 $: spatial decay of the velocity
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 3
SP - 299
EP - 324
AB - We consider the homogeneous time-dependent Oseen system in the whole space $ \mathbb {R}^3 $. The initial data is assumed to behave as $O(|x|^{-1- \epsilon })$, and its gradient as $O(|x|^{-3/2- \epsilon })$, when $|x|$ tends to infinity, where $\epsilon $ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $|u(x,t)|=O(|x|^{-1})$, and its spatial gradient $\nabla _xu$ decreases with the rate $|x|^{-3/2}$, for $|x|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible in the evolutionary case considered in this article. We also treat the case $\epsilon =0$. Then the preceding decay rates of $u$ remain valid, but they are no longer uniform with respect to $t$.
LA - eng
KW - Cauchy problem; time-dependent Oseen system; spatial decay; wake; Cauchy problem; time-dependent Oseen system; spatial decay; wake
UR - http://eudml.org/doc/260628
ER -

References

top
  1. Babenko, K. I., Vasil'ev, M. M., 10.1016/0021-8928(73)90115-9, J. Appl. Math. Mech. 37 (1973), 651-665 translation from Prikl. Mat. Mekh. 37 690-705 (1973), Russian. (1973) Zbl0295.76015MR0347214DOI10.1016/0021-8928(73)90115-9
  2. Bae, H.-O., Jin, B. J., 10.3934/dcdsb.2008.10.1, Discrete Contin. Dyn. Syst., Ser. B 10 (2008), 1-18. (2008) Zbl1141.76021MR2399419DOI10.3934/dcdsb.2008.10.1
  3. Bae, H.-O., Roh, J., 10.1007/s00021-010-0040-z, J. Math. Fluid Mech. 14 (2012), 117-139. (2012) MR2891194DOI10.1007/s00021-010-0040-z
  4. Deuring, P., Exterior stationary Navier-Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient, IASME Trans. 2 (2005), 900-904. (2005) MR2215245
  5. Deuring, P., The single-layer potential associated with the time-dependent Oseen system, In: Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics Chalkida, Greece, May 11-13 (2006), 117-125. (2006) MR2227964
  6. Deuring, P., On volume potentials related to the time-dependent Oseen system, WSEAS Trans. Math. 5 (2006), 252-259. (2006) MR2227964
  7. Deuring, P., On boundary-driven time-dependent Oseen flows, Rencławowicz, Joanna et al. Parabolic and Navier-Stokes Equations. Part 1. Polish Acad. Sci. Inst. Math., Banach Center Publications 81, Warsaw (2008), 119-132. (2008) Zbl1148.76016MR2549327
  8. Deuring, P., A potential theoretic approach to the time-dependent Oseen system, R. Rannacher, A. Sequeira Advances in Mathematical Fluid Mechanics Dedicated to Giovanni Paolo Galdi on the occasion of his 60th birthday. Selected papers of the international conference on mathematical fluid mechanics, Estoril, Portugal, May 21-25, 2007. Springer, Berlin (2010), 191-214. (2010) MR2665032
  9. Deuring, P., 10.1137/080723831, SIAM J. Math. Anal. 41 (2009), 886-922. (2009) Zbl1189.35222MR2529951DOI10.1137/080723831
  10. Deuring, P., A representation formula for the velocity part of 3D time-dependent Oseen flows, (to appear) in J. Math. Fluid Mech. 
  11. Deuring, P., 10.3934/dcds.2013.33.2757, Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 2757-2776. (2013) MR3007725DOI10.3934/dcds.2013.33.2757
  12. Deuring, P., 10.1137/120872255, SIAM J. Math. Anal. 45 (2013), 1388-1421. (2013) MR3056750DOI10.1137/120872255
  13. Deuring, P., Kračmar, S., 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. (2004) Zbl1050.35067MR2074775
  14. Enomoto, Y., Shibata, Y., Local energy decay of solutions to the Oseen equation in the exterior domains, Indiana Univ. Math. J. 53 (2004), 1291-1330. (2004) Zbl1088.35048MR2104279
  15. Enomoto, Y., Shibata, Y., 10.1007/s00021-004-0132-8, J. Math. Fluid Mech. 7 (2005), 339-367. (2005) Zbl1094.35097MR2166980DOI10.1007/s00021-004-0132-8
  16. Farwig, R., 10.1007/BF02571437, Math. Z. 211 (1992), 409-447. (1992) MR1190220DOI10.1007/BF02571437
  17. Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs N.J. (1964). (1964) Zbl0144.34903MR0181836
  18. Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy 39. Springer, New York (1994). (1994) Zbl0949.35005MR1284206
  19. Heywood, J. G., 10.1007/BF02392212, Acta Math. 129 (1972), 11-34. (1972) Zbl0237.35074MR0609550DOI10.1007/BF02392212
  20. Heywood, J. G., 10.1512/iumj.1980.29.29048, Indiana Univ. Math. J. 29 (1980), 639-681. (1980) Zbl0494.35077MR0589434DOI10.1512/iumj.1980.29.29048
  21. Knightly, G. H., 10.1137/0503048, SIAM J. Math. Anal. 3 (1972), 506-511. (1972) MR0312093DOI10.1137/0503048
  22. Knightly, G. H., 10.1007/BFb0086913, R. Rautmann Approximation Methods for Navier-Stokes Problems Proc. Sympos., Univ. Paderborn 1979, Lecture Notes in Math. 771 287-289 Springer, Berlin (1980). (1980) Zbl0458.35082MR0566003DOI10.1007/BFb0086913
  23. Kobayashi, T., Shibata, Y., 10.1007/s002080050134, Math. Ann. 310 (1998), 1-45. (1998) MR1600022DOI10.1007/s002080050134
  24. Kračmar, S., Novotný, A., Pokorný, M., 10.2969/jmsj/05310059, J. Math. Soc. Japan 53 (2001), 59-111. (2001) Zbl0988.76021MR1800524DOI10.2969/jmsj/05310059
  25. Masuda, K., 10.2969/jmsj/02720294, J. Math. Soc. Japan 27 (1975), 294-327. (1975) Zbl0303.76011MR0440224DOI10.2969/jmsj/02720294
  26. Miyakawa, T., 10.32917/hmj/1206133879, Hiroshima Math. J. 12 (1982), 115-140. (1982) Zbl0486.35067MR0647234DOI10.32917/hmj/1206133879
  27. Mizumachi, R., 10.2969/jmsj/03630497, J. Math. Soc. Japan 36 (1984), 497-522. (1984) Zbl0578.76026MR0746708DOI10.2969/jmsj/03630497
  28. Shibata, Y., 10.1090/qam/1672187, Q. Appl. Math. 57 (1999), 117-155. (1999) Zbl1157.35451MR1672187DOI10.1090/qam/1672187
  29. Solonnikov, V. A., 10.1090/trans2/065/03, Am. Math. Soc., Transl., II. Ser. 65 (1967), 51-137. (1967) Zbl0179.42901DOI10.1090/trans2/065/03
  30. Solonnikov, V. A., Estimates for solutions of nonstationary Navier-Stokes equations, Russian Boundary value problems of mathematical physics and related questions in the theory of functions 7. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153-231 English translation, J. Sov. Math. 8 (1977), 467-529. (1977) Zbl0404.35081MR0415097
  31. Takahashi, S., 10.1016/S0362-546X(98)00070-4, Nonlinear Anal., Theory Methods Appl. 37 (1999), 751-789. (1999) Zbl0941.35066MR1692815DOI10.1016/S0362-546X(98)00070-4

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.