| Title:
|
The boundary regularity of a weak solution of the Navier-Stokes equation and its connection to the interior regularity of pressure (English) |
| Author:
|
Neustupa, Jiří |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
6 |
| Year:
|
2003 |
| Pages:
|
547-558 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 ${\mathbb{v}}$. (English) |
| Keyword:
|
Navier-Stokes equations |
| Keyword:
|
regularity |
| MSC:
|
35B65 |
| MSC:
|
35D10 |
| MSC:
|
35Q30 |
| MSC:
|
76D03 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 1099.35087 |
| idMR:
|
MR2025963 |
| DOI:
|
10.1023/B:APOM.0000024493.04008.06 |
| . |
| Date available:
|
2009-09-22T18:15:57Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134550 |
| . |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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. Zbl 1108.35133, MR 1798753 |
| Reference:
|
[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 |
| Reference:
|
[5] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations.Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986. MR 0851383 |
| Reference:
|
[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. |
| Reference:
|
[7] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow.Gordon and Breach, London, 1969. Zbl 0184.52603, MR 0254401 |
| Reference:
|
[8] J. L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications.Springer-Verlag, Berlin-Heidelberg-New York, 1972. |
| Reference:
|
[9] P. L. Lions: Mathematical Topics in Fluid Mechanics, Vol. 1.Clarendon Press, Oxford, 1996. Zbl 0866.76002, MR 1422251 |
| Reference:
|
[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 |
| Reference:
|
[11] J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations.Arch. Rat. Mech. Anal. 9 (1962), 187–195. Zbl 0106.18302, MR 0136885, 10.1007/BF00253344 |
| Reference:
|
[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 |
| Reference:
|
[13] H. Sohr, W. von Wahl: On the regularity of pressure of weak solutions of Navier-Stokes equations.Arch. Math. 46 (1986), 428–439. MR 0847086, 10.1007/BF01210782 |
| Reference:
|
[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. Zbl 0752.35050, MR 1151263, 10.1080/03605309208820841 |
| . |
