Browse > Article
http://dx.doi.org/10.4134/BKMS.2007.44.3.547

TEMPORAL AND SPATIAL DECAY RATES OF NAVIER-STOKES SOLUTIONS IN EXTERIOR DOMAINS  

Bae, Hyeong-Ohk (DEPARTMENT OF NATURAL SCIENCES AJOU UNIVERSITY)
Jin, Bum-Ja (DIVISION OF MATHEMATICS COLLEGE OF SCIENCE MOKPO NATIONAL UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.44, no.3, 2007 , pp. 547-567 More about this Journal
Abstract
We obtain spatial-temporal decay rates of weak solutions of incompressible flows in exterior domains. When a domain has a boundary, the pressure term yields difficulties since we do not have enough information on the pressure term near the boundary. For our calculations we provide an idea which does not require any pressure information. We also estimated the spatial and temporal asymptotic behavior for strong solutions.
Keywords
Navier-Stokes equations; temporal decay; spatial decay; weak solution; strong solution; exterior domain;
Citations & Related Records

Times Cited By Web Of Science : 16  (Related Records In Web of Science)
Times Cited By SCOPUS : 14
연도 인용수 순위
1 H.-O. Bae and H. J. Choe, Decay rate for the incompressible flow in half spaces, Math. Z. 238 (2001), no. 4, 799-816   DOI
2 C. He and Z. Xin, Weighted estimates for nonstationary Navier-Stokes equations in exterior domains, Methods Appl. Anal. 7 (2000), no. 3, 443-458
3 W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flow in exterior domains, Acta Math. 165 (1990), no. 3-4, 189-227   DOI
4 G. A. Seregin, Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech. 4 (2002), no. 1, 1-29   DOI
5 H.-O. Bae and B. J. Jin, Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations 209 (2005), no. 2, 365-391   DOI   ScienceOn
6 H.-O. Bae, Temporal decays in $L^1$ and $L^{\infty}$ for the Stokes flow, J. Differential Equations 222 (2006), no. 1, 1-20   DOI   ScienceOn
7 H.-O. Bae, Temporal and spatial decays for the Stokes flow, submitted
8 H.-O. Bae, Analyticity and asymptotics for the Stokes solutions in a weighted space, J. Math. Anal. Appl. 269 (2002), no. 1, 149-171   DOI   ScienceOn
9 H.-O. Bae and B. J. Jin, Temporal and spatial decays for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), no. 3, 461-477
10 Y. Fujigaki and T. Miyakawa, Asymptotic pro?les of nonstationary incompressible Navier-Stokes flow in the whole space, SIAM J. Math. Anal. 33 (2001), no. 3, 523-544   DOI   ScienceOn
11 T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier- Stokes flow in $R^n$, Funkcial. Ekvac. 43 (2000), no. 3, 541-557
12 R. Farwig and H. Sohr, Global estimates in weighted spaces of weak solutions of the Navier-Stokes equations in exterior domains, Arch. Math. (Basel) 67 (1996), no. 4, 319-330   DOI
13 W. Borchers and T. Miyakawa, Algebraic $L^2$ decay for Navier-Stokes flow in exterior domains. II, Hiroshima Math. J. 21 (1991), no. 3, 621-640
14 L. Brandolese, Space-time decay of Navier-Stokes flow invariant under rotations, Math. Ann. 329 (2004), no. 4, 685-706
15 L. Caffarelli, J. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771-831   DOI
16 R. Farwig and H. Sohr, Weighted energy inequalities for the Navier-Stokes equations in exterior domains, Appl. Anal. 58 (1995), no. 1-2, 157-173   DOI
17 G. P. Galdi and P. Maremonti, Monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the Navier-Stokes equations in exterior domains, Arch. Rational Mech. Anal. 94 (1986), no. 3, 253-266   DOI
18 C. He, Weighted estimates for nonstationary Navier-Stokes equations, J. Differential Equations 148 (1998), no. 2, 422-444   DOI   ScienceOn
19 C. He and T. Miyakawa, On $L^1$-summability and asymptotic profiles for smooth solutions to Navier-Stokes equations in a 3D exterior domain, Math. Z. 245 (2003), no. 2, 387-417   DOI
20 C. He and Z. Xin, On the decay properties of solutions to the non-stationary Navier-Stokes equations in $R^3$, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 3, 597-619
21 E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231
22 O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris 1969
23 H. Iwashita, $L_q-L_p$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in $L_q$ spaces, Math. Ann. 285 (1989), no. 2, 265-288   DOI
24 H. Kozono, Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains, Math. Ann. 320 (2001), no. 4, 709-730   DOI
25 H. Kozono, T. Ogawa, and H. Sohr, Asymptotic behaviour in $L^r$ for weak solutions of the Navier-Stokes equations in exterior domains, Manuscripta Math. 74 (1992), no. 3, 253-275   DOI
26 J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248   DOI
27 T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior do- main, Hiroshima Math. J. 12 (1982), no. 1, 115-140
28 T. Miyakawa and M. E. Schonbek, On optimal decay rates for weak solutions to the Navier-Stokes equations in $R^n$, Proceedings of Partial Differential Equations and Applications (olomouc, 1999), Math. Bohem. 126 (2001), no. 2, 443-455
29 V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), no. 2, 535-552   DOI
30 M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (1986), no. 7, 733-763   DOI
31 S. Takahashi, A weighted equation approach to decay rate estimates for the Navier- Stokes equations, Nonlinear Anal. 37 (1999), no. 6, Ser. A: Theory Methods, 751-789   DOI   ScienceOn
32 M. Wiegner, Decay estimates for strong solutions of the Navier-Stokes equations in ex- terior domains, Navier-Stokes equations and related nonlinear problems (Ferrara, 1999). Ann. Univ. Ferrara Sez. VII (N.S.) 46 (2000), 61-79