October  2014, 7(5): 887-900. doi: 10.3934/dcdss.2014.7.887

Some uniqueness result of the Stokes flow in a half space in a space of bounded functions

1. 

Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

Received  March 2013 Revised  October 2013 Published  May 2014

This paper presents a uniqueness theorem for the Stokes equations in a half space in a space of bounded functions. The Stokes equations is well understood for decaying velocity as $|x|\to\infty$, but less known for non-decaying velocity even for a half space. This paper presents a uniqueness theorem on $L^{\infty}(\mathbb{R}_+^n)$ for unbounded velocity as $t\downarrow 0$. Under suitable sup-bounds both for velocity and pressure gradient, a uniqueness theorem for non-decaying velocity is proved.
Citation: Ken Abe. Some uniqueness result of the Stokes flow in a half space in a space of bounded functions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 887-900. doi: 10.3934/dcdss.2014.7.887
References:
[1]

K. Abe, The Stokes Semigroup on Non-Decaying Spaces,, Ph.D thesis, (2013).   Google Scholar

[2]

K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions,, Acta Math., 211 (2013), 1.  doi: 10.1007/s11511-013-0098-6.  Google Scholar

[3]

K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains,, J. Evol. Equ., 14 (2014), 1.  doi: 10.1007/s00028-013-0197-z.  Google Scholar

[4]

K. Abe, Y. Giga and M. Hieber, Stokes Resolvent Estimates in Spaces of Bounded Functions,, Hokkaido University Preprint Series in Mathematics, (2012).   Google Scholar

[5]

H.-O. Bae and B. Jin, Existence of strong mild solution of the Navier-Stokes equations in the half space with nondecaying initial data,, J. Korean Math. Soc., 49 (2012), 113.  doi: 10.4134/JKMS.2012.49.1.113.  Google Scholar

[6]

W. Desch, M. Hieber and J. Prüss, $L^p$-theory of the Stokes equation in a half space,, J. Evol. Equ., 1 (2001), 115.  doi: 10.1007/PL00001362.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[8]

Y. Giga, S. Matsui and Y. Shimizu, On estimates in Hardy spaces for the Stokes flow in a half space,, Math. Z., 231 (1999), 383.  doi: 10.1007/PL00004735.  Google Scholar

[9]

G. de Rham, Differentiable Manifolds,, Springer-Verlag, (1984).  doi: 10.1007/978-3-642-61752-2.  Google Scholar

[10]

J. Saal, The Stokes operator with Robin boundary conditions in solenoidal subspaces of $L^1(\mathbbR^n_+)$ and $L^\infty(\mathbbR^n_+)$,, Communications in Partial Differential Equations, 32 (2007), 343.  doi: 10.1080/03605300601160408.  Google Scholar

[11]

C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains,, in Mathematical problems relating to the Navier-Stokes equation, (1992), 1.   Google Scholar

[12]

V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity,, Function theory and applications, 114 (2003), 1726.  doi: 10.1023/A:1022317029111.  Google Scholar

[13]

V. A. Solonnikov, Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator,, Russian, 58 (2003), 123.  doi: 10.1070/RM2003v058n02ABEH000613.  Google Scholar

[14]

S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$,, Comm. Pure Appl. Math., 40 (1987), 611.  doi: 10.1002/cpa.3160400506.  Google Scholar

show all references

References:
[1]

K. Abe, The Stokes Semigroup on Non-Decaying Spaces,, Ph.D thesis, (2013).   Google Scholar

[2]

K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions,, Acta Math., 211 (2013), 1.  doi: 10.1007/s11511-013-0098-6.  Google Scholar

[3]

K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains,, J. Evol. Equ., 14 (2014), 1.  doi: 10.1007/s00028-013-0197-z.  Google Scholar

[4]

K. Abe, Y. Giga and M. Hieber, Stokes Resolvent Estimates in Spaces of Bounded Functions,, Hokkaido University Preprint Series in Mathematics, (2012).   Google Scholar

[5]

H.-O. Bae and B. Jin, Existence of strong mild solution of the Navier-Stokes equations in the half space with nondecaying initial data,, J. Korean Math. Soc., 49 (2012), 113.  doi: 10.4134/JKMS.2012.49.1.113.  Google Scholar

[6]

W. Desch, M. Hieber and J. Prüss, $L^p$-theory of the Stokes equation in a half space,, J. Evol. Equ., 1 (2001), 115.  doi: 10.1007/PL00001362.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[8]

Y. Giga, S. Matsui and Y. Shimizu, On estimates in Hardy spaces for the Stokes flow in a half space,, Math. Z., 231 (1999), 383.  doi: 10.1007/PL00004735.  Google Scholar

[9]

G. de Rham, Differentiable Manifolds,, Springer-Verlag, (1984).  doi: 10.1007/978-3-642-61752-2.  Google Scholar

[10]

J. Saal, The Stokes operator with Robin boundary conditions in solenoidal subspaces of $L^1(\mathbbR^n_+)$ and $L^\infty(\mathbbR^n_+)$,, Communications in Partial Differential Equations, 32 (2007), 343.  doi: 10.1080/03605300601160408.  Google Scholar

[11]

C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains,, in Mathematical problems relating to the Navier-Stokes equation, (1992), 1.   Google Scholar

[12]

V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity,, Function theory and applications, 114 (2003), 1726.  doi: 10.1023/A:1022317029111.  Google Scholar

[13]

V. A. Solonnikov, Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator,, Russian, 58 (2003), 123.  doi: 10.1070/RM2003v058n02ABEH000613.  Google Scholar

[14]

S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$,, Comm. Pure Appl. Math., 40 (1987), 611.  doi: 10.1002/cpa.3160400506.  Google Scholar

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