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 and 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, the University of Tokyo, 2013.

[2]

K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Math., 211 (2013), 1-46. doi: 10.1007/s11511-013-0098-6.

[3]

K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains, J. Evol. Equ., 14 (2014), 1-28. doi: 10.1007/s00028-013-0197-z.

[4]

K. Abe, Y. Giga and M. Hieber, Stokes Resolvent Estimates in Spaces of Bounded Functions, Hokkaido University Preprint Series in Mathematics, 2012, no. 1022. Available from: http://eprints3.math.sci.hokudai.ac.jp/2238/.

[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), 113-138. doi: 10.4134/JKMS.2012.49.1.113.

[6]

W. Desch, M. Hieber and J. Prüss, $L^p$-theory of the Stokes equation in a half space, J. Evol. Equ., 1 (2001), 115-142. doi: 10.1007/PL00001362.

[7]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[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), 383-396. doi: 10.1007/PL00004735.

[9]

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

[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), 343-373. doi: 10.1080/03605300601160408.

[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, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992, 1-35.

[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, J. Math. Sci. (N. Y.), 114 (2003), 1726-1740. doi: 10.1023/A:1022317029111.

[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, with Russian summary, Uspekhi Mat. Nauk, 58 (2003), 123-156; translation in Russian Math. Surveys, 58 (2003), 331-365. doi: 10.1070/RM2003v058n02ABEH000613.

[14]

S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$, Comm. Pure Appl. Math., 40 (1987), 611-621. doi: 10.1002/cpa.3160400506.

show all references

References:
[1]

K. Abe, The Stokes Semigroup on Non-Decaying Spaces, Ph.D thesis, the University of Tokyo, 2013.

[2]

K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Math., 211 (2013), 1-46. doi: 10.1007/s11511-013-0098-6.

[3]

K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains, J. Evol. Equ., 14 (2014), 1-28. doi: 10.1007/s00028-013-0197-z.

[4]

K. Abe, Y. Giga and M. Hieber, Stokes Resolvent Estimates in Spaces of Bounded Functions, Hokkaido University Preprint Series in Mathematics, 2012, no. 1022. Available from: http://eprints3.math.sci.hokudai.ac.jp/2238/.

[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), 113-138. doi: 10.4134/JKMS.2012.49.1.113.

[6]

W. Desch, M. Hieber and J. Prüss, $L^p$-theory of the Stokes equation in a half space, J. Evol. Equ., 1 (2001), 115-142. doi: 10.1007/PL00001362.

[7]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[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), 383-396. doi: 10.1007/PL00004735.

[9]

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

[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), 343-373. doi: 10.1080/03605300601160408.

[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, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992, 1-35.

[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, J. Math. Sci. (N. Y.), 114 (2003), 1726-1740. doi: 10.1023/A:1022317029111.

[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, with Russian summary, Uspekhi Mat. Nauk, 58 (2003), 123-156; translation in Russian Math. Surveys, 58 (2003), 331-365. doi: 10.1070/RM2003v058n02ABEH000613.

[14]

S. Ukai, A solution formula for the Stokes equation in $\mathbbR^n_+$, Comm. Pure Appl. Math., 40 (1987), 611-621. doi: 10.1002/cpa.3160400506.

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