Linearized Navier-Stokes equations in \mathbb{R}^3: An approach in weighted Sobolev spaces

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October 2014, 7(5): 901-916. doi: 10.3934/dcdss.2014.7.901

Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces

Chérif Amrouche ^1, , Mohamed Meslameni ^1, and Šárka Nečasová ^2,

1.	Laboratoire de Mathématiques et de leurs Applications, CNRS UMR 5142, Université de Pau et des Pays de l'Adour, 64013 Pau, France, France
2.	Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

Received April 2013 Published May 2014

Abstract
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In this work, we study the linearized Navier-Stokes equations in $\mathbb{R}^3$, the Oseen equations. We are interested in the existence and the uniqueness of generalized and strong solutions in $L^p$-theory which makes analysis more difficult. Our approach rests on the use of weighted Sobolev spaces.

Keywords: the existence, Generalized Oseen equations, weighted Sobolev spaces, generalized solutions, strong solutions., the uniqueness.

Mathematics Subject Classification: 35Q30, 76D03, 76D05, 76D0.

Citation: Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901

References:

[1]	F. Alliot and C. Amrouche, The Stokes problem in $\mathbbR^n$: An approach in weighted Sobolev spaces, Math. Mod. Meth. Appl. Sci., 9 (1999), 723-754. doi: 10.1142/S0218202599000361.
[2]	C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$, Communications in Mathematical Analysis, 10 (2011), 5-29.
[3]	C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for the laplace equation in $\mathbbR^n$, J. Math. Pures et Appl., 73 (1994), 579-606.
[4]	C. Amrouche and M. A. Rodriguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Archive for Rational Mechanics and Analysis, 199 (2011), 597-651. doi: 10.1007/s00205-010-0340-8.
[5]	M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$, Indiana Univ. Math. J., 24 (1975), 897-902.
[6]	R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z, 211 (1992), 409-447. doi: 10.1007/BF02571437.
[7]	R. Farwig, The stationary Navier-Stokes equations in a 3D-exterior domain, in Recent Topics on Mathematical Theory of Viscous Incompressible Fluid (Tsukuba, 1996), Lecture Notes Numer. Appl. Anal., 16, Kinokuniya, Tokyo, 1998, 53-115.
[8]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994.
[9]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994.
[10]	B. Hanouzet, Espace de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272.

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References:

[1]	F. Alliot and C. Amrouche, The Stokes problem in $\mathbbR^n$: An approach in weighted Sobolev spaces, Math. Mod. Meth. Appl. Sci., 9 (1999), 723-754. doi: 10.1142/S0218202599000361.
[2]	C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$, Communications in Mathematical Analysis, 10 (2011), 5-29.
[3]	C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for the laplace equation in $\mathbbR^n$, J. Math. Pures et Appl., 73 (1994), 579-606.
[4]	C. Amrouche and M. A. Rodriguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Archive for Rational Mechanics and Analysis, 199 (2011), 597-651. doi: 10.1007/s00205-010-0340-8.
[5]	M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$, Indiana Univ. Math. J., 24 (1975), 897-902.
[6]	R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z, 211 (1992), 409-447. doi: 10.1007/BF02571437.
[7]	R. Farwig, The stationary Navier-Stokes equations in a 3D-exterior domain, in Recent Topics on Mathematical Theory of Viscous Incompressible Fluid (Tsukuba, 1996), Lecture Notes Numer. Appl. Anal., 16, Kinokuniya, Tokyo, 1998, 53-115.
[8]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994.
[9]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, 1994.
[10]	B. Hanouzet, Espace de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227-272.

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