# American Institute of Mathematical Sciences

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

 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

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 $1< p <\infty$.
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), 109-140.  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), 597-651. 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), 581-607. 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), 263-318.  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), 109-140.  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), 597-651. 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), 581-607. 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), 263-318.  Google Scholar
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