American Institute of Mathematical Sciences

Advanced
November  2010, 14(4): 1537-1564. doi: 10.3934/dcdsb.2010.14.1537

Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition

Jean-Pierre Raymond 1,

1. 

Université de Toulouse & CNRS, Institut de Mathématiques, UMR 5219, 31062 Toulouse Cedex 9

Received  September 2009 Revised  February 2010 Published  August 2010

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with a nonhomogeneous divergence condition. We also prove the existence of global weak solutions to the 3D Navier-Stokes equations when the divergence is not equal to zero. These equations intervene in control problems for the Navier-Stokes equations and in fluid-structure interaction problems.
Keywords: nonhomogeneous divergence condition, nonhomogeneous Dirichlet boundary condition., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q30, 76D05, 76D03; Secondary: 76D0.
Citation: Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[1]

Shijin Ding, Zhilin Lin, Dongjuan Niu. Boundary layer for 3D plane parallel channel flows of nonhomogeneous incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4579-4596. doi: 10.3934/dcds.2020193
[2]

Alain Miranville, Xiaoming Wang. Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 1996, 2 (1) : 95-110. doi: 10.3934/dcds.1996.2.95
[3]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153
[4]

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 and Continuous Dynamical Systems, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353
[5]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[6]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246
[7]

Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4421-4460. doi: 10.3934/dcds.2021042
[8]

Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[9]

Anis Dhifaoui. $ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1403-1420. doi: 10.3934/dcdss.2022086
[10]

Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[11]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[12]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[13]

Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[14]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
[15]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[16]

Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889
[17]

Larissa V. Fardigola. Transformation operators in controllability problems for the wave equations with variable coefficients on a half-axis controlled by the Dirichlet boundary condition. Mathematical Control and Related Fields, 2015, 5 (1) : 31-53. doi: 10.3934/mcrf.2015.5.31
[18]

Lan Zeng, Guoxi Ni, Yingying Li. Low Mach number limit of strong solutions for 3-D full compressible MHD equations with Dirichlet boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5503-5522. doi: 10.3934/dcdsb.2019068
[19]

H. Beirão da Veiga. Vorticity and regularity for flows under the Navier boundary condition. Communications on Pure and Applied Analysis, 2006, 5 (4) : 907-918. doi: 10.3934/cpaa.2006.5.907
[20]

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

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy