American Institute of Mathematical Sciences

Advanced
October  2013, 6(5): 1427-1455. doi: 10.3934/dcdss.2013.6.1427

Long time existence of regular solutions to non-homogeneous Navier-Stokes equations

Wojciech M. Zajączkowski 1,

1. 

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

Received  December 2011 Revised  April 2012 Published  March 2013

We consider the motion of incompressible viscous non-homogene-ous fluid described by the Navier-Stokes equations in a bounded cylinder $\Omega$ under boundary slip conditions. Assume that the $x_3$-axis is the axis of the cylinder. Let $\varrho$ be the density of the fluid, $v$ -- the velocity and $f$ the external force field. Assuming that quantities $\nabla\varrho(0)$, $\partial_{x_3}v(0)$, $\partial_{x_3}f$, $f_3|_{\partial\Omega}$ are sufficiently small in some norms we prove large time regular solutions such that $v\in H^{2+s,1+s/2}(\Omega\times(0,T))$, $\nabla p\in H^{s,s/2}(\Omega\times(0,T))$, $½ < s < 1$ without any restriction on the existence time $T$. The proof is divided into two parts. First an a priori estimate is shown. Next the existence follows from the Leray-Schauder fixed point theorem.
Keywords: regular solutions., Navier-Stokes equations, non-homogeneous fluids, slip boundary conditions, global existence.
Mathematics Subject Classification: Primary: 35A01, 35Q30; Secondary: 76D05, 76D03, 35Q3.
Citation: Wojciech M. Zajączkowski. Long time existence of regular solutions to non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1427-1455. doi: 10.3934/dcdss.2013.6.1427
References:
[1]

S. N. Antontzev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Problems for Mechanics of Nonhomogeneous Fluids,", (in Russian), (1983).   Google Scholar

[2]

O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, "Integral Representation of Functions, and Embedding Theorems,", (in Russian), (1975).   Google Scholar

[3]

M. Burnat and W. M. Zajączkowski, On local motion of a compressible barotropic viscous fluid with the boundary slip condition,, Top. Meth. Nonlin. Anal., 10 (1997), 195.   Google Scholar

[4]

R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space,, J. Funct. Anal., 256 (2009), 881.  doi: 10.1016/j.jfa.2008.11.019.  Google Scholar

[5]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, 3 (1996).   Google Scholar

[6]

B. Nowakowski and W. M. Zajączkowski, Global existence of solutions to Navier-Stokes equations in cylindrical domains,, Appl. Math., 36 (2009), 169.  doi: 10.4064/am36-2-5.  Google Scholar

[7]

B. Nowakowski and W. M. Zajączkowski, Global attractor for Navier-Stokes equaitons in cylindrical domains,, Appl. Math., 36 (2009), 183.  doi: 10.4064/am36-2-6.  Google Scholar

[8]

J. Rencławowicz and W. M. Zajączkowski, Large time regular solutions to the Navier-Stokes equations in cylindrical domains,, Top. Meth. Nonlin. Anal., 32 (2008), 69.   Google Scholar

[9]

W. M. Zajączkowski, Global existence of axially symmetric solutions of incompressible Navier-Stokes equations with large angular component of velocity,, Colloq. Math., 100 (2004), 243.  doi: 10.4064/cm100-2-7.  Google Scholar

[10]

W. M. Zajączkowski, Long time existence of regular solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions,, Studia Math., 169 (2005), 243.  doi: 10.4064/sm169-3-3.  Google Scholar

[11]

W. M. Zajączkowski, Nonstationary Stokes system in Sobolev-Slobodetski spaces,, Math. Ann., (2013).   Google Scholar

[12]

W. M. Zajączkowski, On global regular solutions to the Navier-Stokes equations in cylindrical domains,, Top. Meth. Nonlin. Anal., 37 (2011), 55.   Google Scholar

[13]

W. M. Zajączkowski, Global special regular solutions to the Navier-Stokes equations in a cylindrical domain without the axis of symmetry,, Top. Meth. Nonlin. Anal., 24 (2004), 69.   Google Scholar

[14]

W. M. Zajączkowski, Global regular solutions to the Navier-Stokes equations in a cylinder,, in, 74 (2006), 235.  doi: 10.4064/bc74-0-15.  Google Scholar

[15]

, E. Zadrzyńska and W. M. Zajączkowski,, Nonstationary Stokes system in anisotropic Sobolev spaces, (2013).   Google Scholar

show all references

References:
[1]

S. N. Antontzev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Problems for Mechanics of Nonhomogeneous Fluids,", (in Russian), (1983).   Google Scholar

[2]

O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, "Integral Representation of Functions, and Embedding Theorems,", (in Russian), (1975).   Google Scholar

[3]

M. Burnat and W. M. Zajączkowski, On local motion of a compressible barotropic viscous fluid with the boundary slip condition,, Top. Meth. Nonlin. Anal., 10 (1997), 195.   Google Scholar

[4]

R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space,, J. Funct. Anal., 256 (2009), 881.  doi: 10.1016/j.jfa.2008.11.019.  Google Scholar

[5]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,", Oxford Lecture Series in Mathematics and its Applications, 3 (1996).   Google Scholar

[6]

B. Nowakowski and W. M. Zajączkowski, Global existence of solutions to Navier-Stokes equations in cylindrical domains,, Appl. Math., 36 (2009), 169.  doi: 10.4064/am36-2-5.  Google Scholar

[7]

B. Nowakowski and W. M. Zajączkowski, Global attractor for Navier-Stokes equaitons in cylindrical domains,, Appl. Math., 36 (2009), 183.  doi: 10.4064/am36-2-6.  Google Scholar

[8]

J. Rencławowicz and W. M. Zajączkowski, Large time regular solutions to the Navier-Stokes equations in cylindrical domains,, Top. Meth. Nonlin. Anal., 32 (2008), 69.   Google Scholar

[9]

W. M. Zajączkowski, Global existence of axially symmetric solutions of incompressible Navier-Stokes equations with large angular component of velocity,, Colloq. Math., 100 (2004), 243.  doi: 10.4064/cm100-2-7.  Google Scholar

[10]

W. M. Zajączkowski, Long time existence of regular solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions,, Studia Math., 169 (2005), 243.  doi: 10.4064/sm169-3-3.  Google Scholar

[11]

W. M. Zajączkowski, Nonstationary Stokes system in Sobolev-Slobodetski spaces,, Math. Ann., (2013).   Google Scholar

[12]

W. M. Zajączkowski, On global regular solutions to the Navier-Stokes equations in cylindrical domains,, Top. Meth. Nonlin. Anal., 37 (2011), 55.   Google Scholar

[13]

W. M. Zajączkowski, Global special regular solutions to the Navier-Stokes equations in a cylindrical domain without the axis of symmetry,, Top. Meth. Nonlin. Anal., 24 (2004), 69.   Google Scholar

[14]

W. M. Zajączkowski, Global regular solutions to the Navier-Stokes equations in a cylinder,, in, 74 (2006), 235.  doi: 10.4064/bc74-0-15.  Google Scholar

[15]

, E. Zadrzyńska and W. M. Zajączkowski,, Nonstationary Stokes system in anisotropic Sobolev spaces, (2013).   Google Scholar

[1]

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
[2]

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
[3]

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
[4]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012
[5]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[6]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[7]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783
[8]

Luigi C. Berselli. An elementary approach to the 3D Navier-Stokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 199-219. doi: 10.3934/dcdss.2010.3.199
[9]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020234
[10]

Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021
[11]

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
[12]

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
[13]

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
[14]

Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045
[15]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[16]

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
[17]

Daoyuan Fang, Bin Han, Matthias Hieber. Local and global existence results for the Navier-Stokes equations in the rotational framework. Communications on Pure & Applied Analysis, 2015, 14 (2) : 609-622. doi: 10.3934/cpaa.2015.14.609
[18]

Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861
[19]

Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547
[20]

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

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences