American Institute of Mathematical Sciences

Advanced
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

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

show all references

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

[1]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[2]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[3]

Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[4]

Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
[5]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[6]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[7]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[8]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[9]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[10]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[11]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[12]

Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325
[13]

Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219
[14]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[15]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[16]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[17]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[18]

Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419
[19]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
[20]

Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences