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


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 < $q{\leq}{\infty}$.


Besov space;half space;Navier-Stokes;Stokes;mild solution


Supported by : National Research Foundation of Korea(NRF)


  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.
  4. W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282 (1988), no. 1, 139-155.
  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.
  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.
  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.
  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.
  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.
  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.
  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.
  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.
  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.
  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
  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.
  23. S. Ukai, A solution formula for the Stokes equations in $\mathbb{R}^n_+$, Comm. Pure Appl. Math. 40 (1987), no. 5, 611-621.