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), Nauka, Novosibirsk, 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), Izdat. "Nauka," Moscow, 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), 195-223.  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), 881-927. 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, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 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), 169-182. 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), 183-194. 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), 69-87.  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), 243-263. 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), 243-285. 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), 55-65.  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), 69-105.  Google Scholar

[14]

W. M. Zajączkowski, Global regular solutions to the Navier-Stokes equations in a cylinder, in "Self-Similar Solutions of Nonlinear PDE," Banach Center Publ., 74, Polish Acad. Sci., Warsaw, (2006), 235-255. 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), Nauka, Novosibirsk, 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), Izdat. "Nauka," Moscow, 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), 195-223.  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), 881-927. 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, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 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), 169-182. 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), 183-194. 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), 69-87.  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), 243-263. 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), 243-285. 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), 55-65.  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), 69-105.  Google Scholar

[14]

W. M. Zajączkowski, Global regular solutions to the Navier-Stokes equations in a cylinder, in "Self-Similar Solutions of Nonlinear PDE," Banach Center Publ., 74, Polish Acad. Sci., Warsaw, (2006), 235-255. 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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