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

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.
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

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