# American Institute of Mathematical Sciences

November  2018, 17(6): 2283-2307. doi: 10.3934/cpaa.2018109

## On pressure stabilization method for nonstationary Navier-Stokes equations

 1 Department of Mathematics, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan 2 Graduate School of Pure and Applied Sciences, University of Tsukuba, Tennodai 1-1-1, Tsukuba, Ibaraki, 305-8571, Japan

* Corresponding author

Received  May 2017 Revised  January 2018 Published  June 2018

Fund Project: The first author was partially supproted by JSPS Grant-in-aid for Scientific Research (C) #15K04946

In this paper, we consider the nonstationary Navier-Stokes equations approximated by the pressure stabilization method. We can obtain the local in time existence theorem for the approximated Navier-Stokes equations. Moreover we can obtain the error estimate between the solution to the usual Navier-Stokes equations and the Navier-Stokes equations approximated by the pressure stabilization method.

Citation: Takayuki Kubo, Ranmaru Matsui. On pressure stabilization method for nonstationary Navier-Stokes equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2283-2307. doi: 10.3934/cpaa.2018109
##### References:
 [1] F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in W. Hackbush, editor, Efficient Solutions of Elliptic Systems, Note on Numerical Fluid Mechanics, Braunschweig, 10 1984.  Google Scholar [2] A. P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math, 4 (1961), 33-49.   Google Scholar [3] R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness Fourier multipliers and problems of elliptic and parabolic type, Memories of the American Mathematical Society, 788 (2003).  Google Scholar [4] Y. Enomoto and Y. Shibata, On the $\mathcal{R}$-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcialaj Ekvacioj, (2013), 441-505.   Google Scholar [5] Y. Enomoto, L.v. Below and Y. Shibata, On some free boundary problem for a compressible barotropic viscous fluid flow, Ann Univ. Ferrara, 60 (2014), 55-89.   Google Scholar [6] G. P. Galdi, An Introduction to The Mathematical Theory of The Navier-Stokes Equations, Vol. Ⅰ: Linear Steady Problems, Vol. Ⅱ: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Springer Verlag New York, 38, 39 (1994), 2nd edition (1998).  Google Scholar [7] S. A. Nazarov and M. Specovius-Neugebauer, Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems, Differential and Integral Equations, 17 (2004), 1359-1394.   Google Scholar [8] A. Prohl, Projection and Quasi-Compressiblility Methods for Solving The Incompressible Navier-Stokes Equations, Advances in Numerical Mathematics, 1997.  Google Scholar [9] Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, Journal of Mathematical Fluid Mechanics, (2013), 1-40.   Google Scholar [10] Y. Shibata and T. Kubo, (Japanease) [Nonlinear partial differential equations] Asakura Shoten, 2012. Google Scholar [11] Y. Shibata and S. Shimizu, On the maximal $L_p-L_q$ regularity of the Stokes problem with first order boundary condition: model problems, The Mathematical Society of Japan, 64 (2012), 561-626.   Google Scholar [12] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math.Ann., 319 (2001), 735-758.   Google Scholar

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##### References:
 [1] F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in W. Hackbush, editor, Efficient Solutions of Elliptic Systems, Note on Numerical Fluid Mechanics, Braunschweig, 10 1984.  Google Scholar [2] A. P. Calderon, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. in Pure Math, 4 (1961), 33-49.   Google Scholar [3] R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness Fourier multipliers and problems of elliptic and parabolic type, Memories of the American Mathematical Society, 788 (2003).  Google Scholar [4] Y. Enomoto and Y. Shibata, On the $\mathcal{R}$-sectoriality and the initial boundary value problem for the viscous compressible fluid flow, Funkcialaj Ekvacioj, (2013), 441-505.   Google Scholar [5] Y. Enomoto, L.v. Below and Y. Shibata, On some free boundary problem for a compressible barotropic viscous fluid flow, Ann Univ. Ferrara, 60 (2014), 55-89.   Google Scholar [6] G. P. Galdi, An Introduction to The Mathematical Theory of The Navier-Stokes Equations, Vol. Ⅰ: Linear Steady Problems, Vol. Ⅱ: Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Springer Verlag New York, 38, 39 (1994), 2nd edition (1998).  Google Scholar [7] S. A. Nazarov and M. Specovius-Neugebauer, Optimal results for the Brezzi-Pitkäranta approximation of viscous flow problems, Differential and Integral Equations, 17 (2004), 1359-1394.   Google Scholar [8] A. Prohl, Projection and Quasi-Compressiblility Methods for Solving The Incompressible Navier-Stokes Equations, Advances in Numerical Mathematics, 1997.  Google Scholar [9] Y. Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, Journal of Mathematical Fluid Mechanics, (2013), 1-40.   Google Scholar [10] Y. Shibata and T. Kubo, (Japanease) [Nonlinear partial differential equations] Asakura Shoten, 2012. Google Scholar [11] Y. Shibata and S. Shimizu, On the maximal $L_p-L_q$ regularity of the Stokes problem with first order boundary condition: model problems, The Mathematical Society of Japan, 64 (2012), 561-626.   Google Scholar [12] L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math.Ann., 319 (2001), 735-758.   Google Scholar
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