# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] W. M. Zajączkowski, Nonstationary Stokes system in Sobolev-Slobodetski spaces, Math. Ann., (2013). [12] W. M. Zajączkowski, On global regular solutions to the Navier-Stokes equations in cylindrical domains, Top. Meth. Nonlin. Anal., 37 (2011), 55-65. [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. [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. [15] , E. Zadrzyńska and W. M. Zajączkowski,, Nonstationary Stokes system in anisotropic Sobolev spaces, (2013).

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##### References:
 [1] S. N. Antontzev, A. V. Kazhikhov and V. N. Monakhov, "Boundary Problems for Mechanics of Nonhomogeneous Fluids," (in Russian), Nauka, Novosibirsk, 1983. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] W. M. Zajączkowski, Nonstationary Stokes system in Sobolev-Slobodetski spaces, Math. Ann., (2013). [12] W. M. Zajączkowski, On global regular solutions to the Navier-Stokes equations in cylindrical domains, Top. Meth. Nonlin. Anal., 37 (2011), 55-65. [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. [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. [15] , E. Zadrzyńska and W. M. Zajączkowski,, Nonstationary Stokes system in anisotropic Sobolev spaces, (2013).
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