L^p-theory for the Navier-Stokes equations with pressure boundary conditions

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October 2013, 6(5): 1113-1137. doi: 10.3934/dcdss.2013.6.1113

$L^p$-theory for the Navier-Stokes equations with pressure boundary conditions

Chérif Amrouche ^1, and Nour El Houda Seloula ^2,

1.	Université de Pau et des Pays de l'Adour, LMA, Avenue de l'Université, 64013 Pau cedex, France
2.	Laboratoire de Mathématiques Nicolas Oresme BP 5186, UMR 6139 CNRS, Université de Caen Basse Normandie 14032 Cedex, France

Received December 2011 Revised April 2012 Published March 2013

Abstract
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We consider the Navier-Stokes equations with pressure boundary conditions in the case of a bounded open set, connected of class $\mathcal{C}^{\,1,1}$ of $\mathbb{R}^3$. We prove existence of solution by using a fixed point theorem over the type-Oseen problem. This result was studied in [5] in the Hilbertian case. In our study we give the $L^p$-theory for $1< p <\infty$.

Keywords: Stokes equations, Navier-Stokes equations, inf-sup condition., Oseen equations, pressure boundary conditions.

Mathematics Subject Classification: Primary: 35J25, 35J47, 35J50, 35J57; Secondary: 76D05, 76D0.

Citation: Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113

References:

[1]	C. Amrouche and V. Girault, Decomposition of vector space and application to the Stokes problem in arbitrary dimension,, Czechoslovak Math. J., 44 (1994), 109. Google Scholar
[2]	C. Amrouche and M. Ángeles Rodríguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data,, Arch. Rational. Mech. Anal., 199 (2011), 597. doi: 10.1007/s00205-010-0340-8. Google Scholar
[3]	C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields. Application to the Stokes equations with pressure boundary condition,, to appear in M3AS., (). Google Scholar
[4]	C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions,, Differential Equations and Applications, 3 (2011), 581. doi: 10.7153/dea-03-36. Google Scholar
[5]	C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure,, Japan. J. Math. (N.S.), 20 (1994), 263. Google Scholar

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References:

[1]	C. Amrouche and V. Girault, Decomposition of vector space and application to the Stokes problem in arbitrary dimension,, Czechoslovak Math. J., 44 (1994), 109. Google Scholar
[2]	C. Amrouche and M. Ángeles Rodríguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data,, Arch. Rational. Mech. Anal., 199 (2011), 597. doi: 10.1007/s00205-010-0340-8. Google Scholar
[3]	C. Amrouche and N. Seloula, $L^p$-theory for vector potentials and Sobolev's inequalities for vector fields. Application to the Stokes equations with pressure boundary condition,, to appear in M3AS., (). Google Scholar
[4]	C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions,, Differential Equations and Applications, 3 (2011), 581. doi: 10.7153/dea-03-36. Google Scholar
[5]	C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure,, Japan. J. Math. (N.S.), 20 (1994), 263. Google Scholar

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