American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 1234-1243. doi: 10.3934/proc.2011.2011.1234

Global regular solutions to the Navier-Stokes equations with large flux

Joanna Rencławowicz 1, and  Wojciech M. Zajączkowski 2,

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland
2. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw

Received  July 2010 Revised  March 2011 Published  October 2011

We consider the Navier-Stokes motion in a bounded cylinder with boundary slip conditions. We assume an inflow and an outflow of the fluid through the bottom and the top of the cylinder where the magnitude of the flux is not restricted. We require that the derivatives of the initial velocity and the external force with respect to the variable along the axis of the cylinder are sufficiently small. Under these conditions we are able to prove global existence of regular solutions. Since we are interested in nonvanishing in time flux we need to use the Hopf function to derive global energy estimate.
Keywords: large flux, Neumann boundary- value problem, Navier-Stokes equation, global solutions, weighted Sobolev spaces, Dirichlet boundary-value problem.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D05.
Citation: Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[1]

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
[2]

John R. Graef, Lingju Kong, Bo Yang. Positive solutions of a nonlinear higher order boundary-value problem. Conference Publications, 2009, 2009 (Special) : 276-285. doi: 10.3934/proc.2009.2009.276
[3]

Hideo Ikeda, Koji Kondo, Hisashi Okamoto, Shoji Yotsutani. On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows. Communications on Pure & Applied Analysis, 2003, 2 (3) : 381-390. doi: 10.3934/cpaa.2003.2.381
[4]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6843-6864. doi: 10.3934/dcds.2019234
[5]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[6]

Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
[7]

Kateryna Marynets. Study of a nonlinear boundary-value problem of geophysical relevance. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4771-4781. doi: 10.3934/dcds.2019194
[8]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741
[9]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[10]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[11]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
[12]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121
[13]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[14]

Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1
[15]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084
[16]

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
[17]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[18]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic & Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[19]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[20]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

 Impact Factor: 

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences