About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Deuring, Paul
The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbb {R}^3 $: spatial decay of the velocity. (English). Mathematica Bohemica, vol. 138 (2013), issue 3, pp. 299-324
MSC: 35B25, 35Q30, 35Q35, 65N30, 76D05 | MR 3136499 | Zbl 06260035 | DOI: 10.21136/MB.2013.143439
Keywords:
Cauchy problem; time-dependent Oseen system; spatial decay; wake
Summary:
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 $t$.
References:
[1] Babenko, K. I., Vasil'ev, M. M.: On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body. J. Appl. Math. Mech. 37 (1973), 651-665 translation from Prikl. Mat. Mekh. 37 690-705 (1973), Russian. DOI 10.1016/0021-8928(73)90115-9 | MR 0347214 | Zbl 0295.76015
[2] Bae, H.-O., Jin, B. J.: Estimates of the wake for the 3D Oseen equations. Discrete Contin. Dyn. Syst., Ser. B 10 (2008), 1-18. DOI 10.3934/dcdsb.2008.10.1 | MR 2399419 | Zbl 1141.76021
[3] Bae, H.-O., Roh, J.: 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 | MR 2891194
[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. MR 2215245
[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. MR 2227964
[6] Deuring, P.: On volume potentials related to the time-dependent Oseen system. WSEAS Trans. Math. 5 (2006), 252-259. MR 2227964
[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. MR 2549327 | Zbl 1148.76016
[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. MR 2665032
[9] Deuring, P.: Spatial decay of time-dependent Oseen flows. SIAM J. Math. Anal. 41 (2009), 886-922. DOI 10.1137/080723831 | MR 2529951 | Zbl 1189.35222
[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.: Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support. Discrete Contin. Dyn. Syst. Ser. A 33 (2013), 2757-2776. DOI 10.3934/dcds.2013.33.2757 | MR 3007725
[12] Deuring, P.: Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity. SIAM J. Math. Anal. 45 (2013), 1388-1421. DOI 10.1137/120872255 | MR 3056750
[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. MR 2074775 | Zbl 1050.35067
[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. MR 2104279 | Zbl 1088.35048
[15] Enomoto, Y., Shibata, Y.: 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 | MR 2166980 | Zbl 1094.35097
[16] Farwig, R.: 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 | MR 1190220
[17] Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs N.J. (1964). MR 0181836 | Zbl 0144.34903
[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). MR 1284206 | Zbl 0949.35005
[19] Heywood, J. G.: The exterior nonstationary problem for the Navier-Stokes equations. Acta Math. 129 (1972), 11-34. DOI 10.1007/BF02392212 | MR 0609550 | Zbl 0237.35074
[20] Heywood, J. G.: 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 | MR 0589434 | Zbl 0494.35077
[21] Knightly, G. H.: A Cauchy problem for the Navier-Stokes equations in $ \mathbb{R} ^n$. SIAM J. Math. Anal. 3 (1972), 506-511. DOI 10.1137/0503048 | MR 0312093
[22] Knightly, G. H.: Some decay properties of solutions of the Navier-Stokes equations. R. Rautmann Approximation Methods for Navier-Stokes Problems Proc. Sympos., Univ. Paderborn 1979, Lecture Notes in Math. 771 287-289 Springer, Berlin (1980). DOI 10.1007/BFb0086913 | MR 0566003 | Zbl 0458.35082
[23] Kobayashi, T., Shibata, Y.: On the Oseen equation in three dimensional exterior domains. Math. Ann. 310 (1998), 1-45. DOI 10.1007/s002080050134 | MR 1600022
[24] Kračmar, S., Novotný, A., Pokorný, M.: Estimates of Oseen kernels in weighted $L^p$ spaces. J. Math. Soc. Japan 53 (2001), 59-111. DOI 10.2969/jmsj/05310059 | MR 1800524 | Zbl 0988.76021
[25] Masuda, K.: On the stability of incompressible viscous fluid motions past objects. J. Math. Soc. Japan 27 (1975), 294-327. DOI 10.2969/jmsj/02720294 | MR 0440224 | Zbl 0303.76011
[26] Miyakawa, T.: On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12 (1982), 115-140. DOI 10.32917/hmj/1206133879 | MR 0647234 | Zbl 0486.35067
[27] Mizumachi, R.: On the asymptotic behavior of incompressible viscous fluid motions past bodies. J. Math. Soc. Japan 36 (1984), 497-522. DOI 10.2969/jmsj/03630497 | MR 0746708 | Zbl 0578.76026
[28] Shibata, Y.: On an exterior initial-boundary value problem for Navier-Stokes equations. Q. Appl. Math. 57 (1999), 117-155. DOI 10.1090/qam/1672187 | MR 1672187 | Zbl 1157.35451
[29] Solonnikov, V. A.: A priori estimates for second order parabolic equations. Am. Math. Soc., Transl., II. Ser. 65 (1967), 51-137. DOI 10.1090/trans2/065/03 | Zbl 0179.42901
[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. MR 0415097 | Zbl 0404.35081
[31] Takahashi, S.: A weighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal., Theory Methods Appl. 37 (1999), 751-789. DOI 10.1016/S0362-546X(98)00070-4 | MR 1692815 | Zbl 0941.35066
Partner of
EuDML logo