EXISTENCE OF STRONG MILD SOLUTION OF THE NAVIER-STOKES EQUATIONS IN THE HALF SPACE WITH NONDECAYING INITIAL DATA

Title & Authors
EXISTENCE OF STRONG MILD SOLUTION OF THE NAVIER-STOKES EQUATIONS IN THE HALF SPACE WITH NONDECAYING INITIAL DATA
Bae, Hyeong-Ohk; Jin, Bum-Ja;

Abstract
We construct a mild solutions of the Navier-Stokes equations in half spaces for nondecaying initial velocities. We also obtain the uniform bound of the velocity field and its derivatives.
Keywords
Navier-Stokes;nondecaying data;existence;uniqueness;
Language
English
Cited by
1.
Weighted Decay Results for the Nonstationary Stokes Flow and Navier–Stokes Equations in Half Spaces, Journal of Mathematical Fluid Mechanics, 2015, 17, 4, 599
2.
Some uniqueness result of the stokes flow in a half space in a space of bounded functions, Discrete and Continuous Dynamical Systems - Series S, 2014, 7, 5, 887
3.
Long-time behavior for the nonstationary Navier–Stokes flows in, Journal of Functional Analysis, 2014, 266, 3, 1511
4.
Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface, Analysis & PDE, 2014, 7, 6, 1253
5.
The L ∞-Stokes semigroup in exterior domains, Journal of Evolution Equations, 2014, 14, 1, 1
6.
Navier wall law for nonstationary viscous incompressible flows, Journal of Differential Equations, 2016, 260, 10, 7358
7.
The Navier–Stokes Equations in a Space of Bounded Functions, Communications in Mathematical Physics, 2015, 338, 2, 849
8.
Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Mathematica, 2013, 211, 1, 1
9.
On estimates for the Stokes flow in a space of bounded functions, Journal of Differential Equations, 2016, 261, 3, 1756
10.
A Liouville Theorem for the Planer Navier-Stokes Equations with the No-Slip Boundary Condition and Its Application to a Geometric Regularity Criterion, Communications in Partial Differential Equations, 2014, 39, 10, 1906
11.
Exterior Navier-Stokes flows for bounded data, Mathematische Nachrichten, 2016
References
1.
H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech. 2 (2000), no. 1, 16-98.

2.
H.-O. Bae, Temporal and spatial decays for the Stokes flow, J. Math. Fluid Mech. 10 (2008), no. 4, 503-530.

3.
H.-O. Bae, Temporal decays in $L^1$ and $L^{\infty}$ for the Stokes flow, J. Differential Equations 222 (2006), no. 1, 1-20.

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.
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. POMI, 318 (2004)

6.
F. Crispo and P. Maremonti, On the (x, t) asymptotic properties of solutions of the Navier-Stokes equations in the half space, J. Math. Sci. 136 (2006), no. 2, 3735-3767.

7.
Y. Fujigaki and T. Miyakawa, Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the half-space, Methods Appl. Anal. 8 (2001), no. 1, 121-157.

8.
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.

9.
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.

10.
Y. Giga, K. Inui, A. Mahalov, A. Matsui, and J. Saal, Rotating Navier-Stokes equations in $R^3_+$ with initial data nondecreasing at infinity: the Ekman boundary layer problem, Arch. Ration. Mech. Anal. 186 (2007), no. 2, 177-224.

11.
M. Hieber and O. Sawada, The Navier-Stokes equations in $R^n$ with linearly growing initial data, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 269-285.

12.
J. Kato, The uniqueness of nondecaying solutions for the Navier-Stokes equations, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 159-175.

13.
P. Maremonti, Stokes and Navier-Stokes problems in a half space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity, Journal of Mathematical Sciences 159 (2009), no. 4, 486-523.

14.
P. Maremonti and G. Starita, Nonstationary Stokes equations in a half-space with con-tinuous initial data, Zap. Nauchn. Sem. POMI, 295

15.
P. Maremonti and G. Starita, Nonstationary Stokes equations in a half-space with con-tinuous initial data, J. Math. Sci. 127 (2005), no. 2, 1886-1914.

16.
E. Nakai and T. Yoneda, Generalized Campanato spaces and the uniqueness of nondecaying solutions for the Navier-Stokes equations, preprint, 2009.

17.
Y. Shimizu, $L^{\infty}$ estimate of the first-order space derivatives of Stokes ow in a half space, Funkcial. Ekvac. 42 (1999), no. 2, 291-309.

18.
V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, Problemy Matematisheskogo Analiza, (2003) no. 25, 189-210

19.
V. A. Solonnikov, 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

20.
S. Ukai, A solution formula for the Stokes equation in $R^n_+$, Comm. Pure Appl. Math. 40 (1987), no. 5, 611-621.