Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Published : 2007.08.31
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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.

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References

  1. H.-O. Bae, Temporal decays in $L^1$ and $L^{\infty}$ for the Stokes flow, J. Differential Equations 222 (2006), no. 1, 1-20 https://doi.org/10.1016/j.jde.2005.01.001
  2. H.-O. Bae, Temporal and spatial decays for the Stokes flow, submitted
  3. H.-O. Bae, Analyticity and asymptotics for the Stokes solutions in a weighted space, J. Math. Anal. Appl. 269 (2002), no. 1, 149-171 https://doi.org/10.1016/S0022-247X(02)00011-2
  4. H.-O. Bae and H. J. Choe, Decay rate for the incompressible flow in half spaces, Math. Z. 238 (2001), no. 4, 799-816 https://doi.org/10.1007/s002090100276
  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 https://doi.org/10.1016/j.jde.2004.09.011
  6. 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
  7. 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 https://doi.org/10.1007/BF02391905
  8. 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
  9. L. Brandolese, Space-time decay of Navier-Stokes flow invariant under rotations, Math. Ann. 329 (2004), no. 4, 685-706
  10. 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 https://doi.org/10.1002/cpa.3160350604
  11. 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 https://doi.org/10.1007/BF01197597
  12. 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 https://doi.org/10.1080/00036819508840368
  13. 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 https://doi.org/10.1137/S0036141000367072
  14. 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 https://doi.org/10.1007/BF00279866
  15. C. He, Weighted estimates for nonstationary Navier-Stokes equations, J. Differential Equations 148 (1998), no. 2, 422-444 https://doi.org/10.1006/jdeq.1998.3472
  16. 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 https://doi.org/10.1007/s00209-003-0551-x
  17. C. He and Z. Xin, Weighted estimates for nonstationary Navier-Stokes equations in exterior domains, Methods Appl. Anal. 7 (2000), no. 3, 443-458
  18. 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
  19. E. Hopf, Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213-231
  20. 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 https://doi.org/10.1007/BF01443518
  21. 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 https://doi.org/10.1007/PL00004492
  22. 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 https://doi.org/10.1007/BF02567671
  23. 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
  24. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), no. 1, 193-248 https://doi.org/10.1007/BF02547354
  25. T. Miyakawa, On space-time decay properties of nonstationary incompressible Navier- Stokes flow in $R^n$, Funkcial. Ekvac. 43 (2000), no. 3, 541-557
  26. T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior do- main, Hiroshima Math. J. 12 (1982), no. 1, 115-140
  27. 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
  28. V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math. 66 (1976), no. 2, 535-552 https://doi.org/10.2140/pjm.1976.66.535
  29. M. E. Schonbek, Large time behaviour of solutions to the Navier-Stokes equations, Comm. Partial Differential Equations 11 (1986), no. 7, 733-763 https://doi.org/10.1080/03605308608820443
  30. 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 https://doi.org/10.1007/s00021-002-8533-z
  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 https://doi.org/10.1016/S0362-546X(98)00070-4
  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

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  29. On Weighted Estimates for the Stokes Flows, with Application to the Navier–Stokes Equations vol.20, pp.3, 2018, https://doi.org/10.1007/s00021-018-0360-y
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