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Navier-Stokes equations; regularity
We assume that ${\mathbb{v}}$ is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of ${\mathbb{v}}$ near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of ${\mathbb{v}}$.
[1] H. Fujita, H. Morimoto: On fractional powers of the Stokes operator. Proc. Japan Acad. Ser. A Math. Sci. 16 (1970), 1141–1143. MR 0296755
[2] G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems. Springer-Verlag, New York-Berlin-Heidelberg, 1994. MR 1284205
[3] G. P. Galdi: An Introduction to the Navier-Stokes initial-boundary value problem. Fundamental Directions in Mathematical Fluid Mechanics, series “Advances in Mathematical Fluid Mechanics”, G. P. Galdi, J. Heywood and R. Rannacher (eds.), Birkhäuser-Verlag, Basel, 2000, pp. 1–98. MR 1798753 | Zbl 1108.35133
[4] Y. Giga: Domains of fractional powers of the Stokes operator in $L^r$ spaces. Arch. Rat. Mech. Anal. 89 (1985), 254–281. MR 0786549
[5] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986. MR 0851383
[6] P. Kučera, Z. Skalák: Smoothness of the derivative of velocity in the vicinity of regular points of the Navier-Stokes equations. Proc. of the 4th seminar “Euler and Navier-Stokes Equations (Theory, Numerical Solution, Applications)”, K. Kozel, J. Příhoda and M. Feistauer (eds.), Institute of Thermomechanics of the Academy of Sciences of the Czech Republic, Prague, 2001, pp. 83–86.
[7] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, London, 1969. MR 0254401 | Zbl 0184.52603
[8] J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, Berlin-Heidelberg-New York, 1972.
[9] P. L. Lions: Mathematical Topics in Fluid Mechanics, Vol. 1. Clarendon Press, Oxford, 1996. MR 1422251 | Zbl 0866.76002
[10] J. Neustupa, P. Penel: Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D  Navier-Stokes equations. Mathematical Fluid Mechanics, Recent Results and Open Problems, series “Advances in Mathematical Fluid Mechanics”, J. Neustupa, P. Penel (eds.), Birkhäuser-Verlag, Basel, 2001, pp. 237–268. MR 1865056
[11] J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9 (1962), 187–195. MR 0136885 | Zbl 0106.18302
[12] Z. Skalák, P. Kučera: Regularity of pressure in the Navier-Stokes equations. Proc. of the Int. Conf. “Mathematical and Computer Modelling in Science and Engineering” dedicated to K. Rektorys, Czech Technical University, Prague, 2003, pp. 27–30. MR 2025965
[13] H. Sohr, W. von Wahl: On the regularity of pressure of weak solutions of Navier-Stokes equations. Arch. Math. 46 (1986), 428–439. DOI 10.1007/BF01210782 | MR 0847086
[14] S. Takahashi: On a regularity criterion up to the boundary for weak solutions of the Navier-Stokes equations. Commun. Partial Differ. Equations 17 (1992), 261–283. DOI 10.1080/03605309208820841 | MR 1151263 | Zbl 0752.35050
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