# American Institute of Mathematical Sciences

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

 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

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.
Citation: Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & 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. doi: 10.1142/S0218202599000361. Google Scholar [2] C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$,, Communications in Mathematical Analysis, 10 (2011), 5. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00205-010-0340-8. Google Scholar [5] M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$,, Indiana Univ. Math. J., 24 (1975), 897. Google Scholar [6] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z, 211 (1992), 409. doi: 10.1007/BF02571437. Google Scholar [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), 53. Google Scholar [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, (1994). Google Scholar [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, (1994). Google Scholar [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. Google Scholar

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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. doi: 10.1142/S0218202599000361. Google Scholar [2] C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$,, Communications in Mathematical Analysis, 10 (2011), 5. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00205-010-0340-8. Google Scholar [5] M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$,, Indiana Univ. Math. J., 24 (1975), 897. Google Scholar [6] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z, 211 (1992), 409. doi: 10.1007/BF02571437. Google Scholar [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), 53. Google Scholar [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, (1994). Google Scholar [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, (1994). Google Scholar [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. Google Scholar
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