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http://dx.doi.org/10.4134/JKMS.2013.50.4.771

NAVIER-STOKES EQUATIONS IN BESOV SPACE B-s∞,∞(ℝn+)  

Jin, Bum Ja (Department of Mathematics Education, Mokpo National University)
Publication Information
Journal of the Korean Mathematical Society / v.50, no.4, 2013 , pp. 771-795 More about this Journal
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 < $q{\leq}{\infty}$.
Keywords
Besov space; half space; Navier-Stokes; Stokes; mild solution;
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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.   DOI
4 W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282 (1988), no. 1, 139-155.   DOI
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.   DOI
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.   DOI   ScienceOn
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.   DOI
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.   DOI
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI   ScienceOn
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.   DOI
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.   DOI
19 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.
20 O. Sawada, On time-local solvability of the Navier-Stokes equations in Besov spaces, Adv. Differential Equations 8 (2003), no. 4, 385-412.
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   DOI
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.   DOI
23 S. Ukai, A solution formula for the Stokes equations in $\mathbb{R}^n_+$, Comm. Pure Appl. Math. 40 (1987), no. 5, 611-621.   DOI