Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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NAVIER-STOKES EQUATIONS IN BESOV SPACE B-s∞,∞(ℝn+)

  • Jin, Bum Ja (Department of Mathematics Education, Mokpo National University)
  • Received : 2011.06.06
  • Published : 2013.07.01
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Abstract

In this paper we consider the Navier-Stokes equations in the half space. Our aim is to construct a mild solution for initial data in $B^{-\alpha}_{{\infty},{\infty}}(\mathbb{R}^n_+)$, 0 < ${\alpha}$ < 1. To do this, we derive the estimate of the Stokes flow with singular initial data in $B^{-\alpha}_{{\infty},q}(\mathbb{R}^n_+)$, 0 < ${\alpha}$ < 1, 1 < $q{\leq}{\infty}$.

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References

  1. R. A. Adams, Sobolev Spaces, Academic press, 1975.
  2. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd Ed., Academic press, 2003.
  3. H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech. 2 (2000), no. 1, 16-98. https://doi.org/10.1007/s000210050018
  4. W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282 (1988), no. 1, 139-155. https://doi.org/10.1007/BF01457017
  5. J. Bergh and J. Lofstrom, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.
  6. M. Cannone, F. Planchon, and M. Schonbek, Strong solutions to the incompressible Navier-Stokes equations in the half space, Comm. Partial Differential Equations 25 (2000), no. 5-6, 903-924. https://doi.org/10.1080/03605300008821536
  7. F. Crispo and P. Maremonti, On the (x, t) asymptotic properties of solutions of the Navier-Stokes equations in the half space, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 36 [35], 147-202, 311; translation in J. Math. Sci. (N. Y.) 136 (2006), no. 2, 3735-3767.
  8. Y. Fujigaki and T. Miyakawa, Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the half-space, Methods Appl. Anal. 8 (2001), no. 1, 121-158.
  9. Y. Giga, K. Inui, J. Kato, and S. Matsui, Remarks on the uniqueness of bounded solu-tions of the Navier-Stokes equations, Nonlinear Anal. 47 (2001), no. 6, 4151-4156. https://doi.org/10.1016/S0362-546X(01)00532-6
  10. Y. Giga, K. Inui, and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Advances in fluid dynamics, 27-68, Quad. Mat., 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999.
  11. Y. Giga, S. Matsui, and O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech. 3 (2001), no. 3, 302-315. https://doi.org/10.1007/PL00000973
  12. Y. Giga, S. Matsui, and Y. Shimizu, On estimates in Hardy spaces for the Stokes flow in a half-space, Math. Z. 231 (1999), no. 2, 383-396. https://doi.org/10.1007/PL00004735
  13. P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, London, 1985.
  14. H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22-35. https://doi.org/10.1006/aima.2000.1937
  15. H. Kozono, T. Ogawa, and Y. Taniuchi, Navier-Stokes equations in the Besov space near $L^{\infty}$ and BMO, Kyushu J. Math. 57 (2003), no. 2, 303-324. https://doi.org/10.2206/kyushujm.57.303
  16. Lemarie-Rieusset and A. Zhioua, Weakly singular initial values for the Stokes equation on a half space, J. Math. Anal. Appl. 320 (2006), no. 1, 205-229. https://doi.org/10.1016/j.jmaa.2005.06.070
  17. P. Maremonti, Stokes and Navier-Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity, J. Math. Sci. (N. Y.) 159 (2009), no. 4, 486-523. https://doi.org/10.1007/s10958-009-9458-3
  18. P. Maremonti and G. Starita, On the nonstationary Stokes equations in half-space with continuous initial data, J. Math. Sci. (N. Y.) 127 (2005), no. 2, 1886-1914. https://doi.org/10.1007/s10958-005-0149-4
  19. O. Sawada, On time-local solvability of the Navier-Stokes equations in Besov spaces, Adv. Differential Equations 8 (2003), no. 4, 385-412.
  20. Y. Shimizu, $L^{\infty}$-estimate of first-order space derivatives of Stokes flow in a half space, Funkcial. Ekvac. 42 (1999), no. 2, 291-309.
  21. V. A. Solonnikov, On estimates for solutions to the nonstationary Stokes problem in anistropic Sobolev spaces and estimates for the resolvent of the Stokes operator, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 123-156; translation in Russian Math. Surveys 58 (2003), no. 2, 331-365 https://doi.org/10.4213/rm613
  22. S. Ukai, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci. (N. Y.) 114 (2003), no. 5, 1726-1740. https://doi.org/10.1023/A:1022317029111
  23. S. Ukai, A solution formula for the Stokes equations in $\mathbb{R}^n_+$, Comm. Pure Appl. Math. 40 (1987), no. 5, 611-621. https://doi.org/10.1002/cpa.3160400506