# Article

Full entry | PDF   (2.5 MB)
Keywords:
non-Newtonian incompressible fluids; Navier-Stokes equations; Cauchy problem
Summary:
We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor $\tau ^V(\mathbb{e}) = \tau (\mathbb{e}) - 2\mu _1 \Delta \mathbb{e}$, where the nonlinear function $\tau (\mathbb{e})$ satisfies $\tau _{ij}(\mathbb{e})e_{ij} \ge c|\mathbb{e}|^p$ or $\tau _{ij}(\mathbb{e})e_{ij} \ge c(|\mathbb{e}|^2+|\mathbb{e}|^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p > 1$ for both models. Then, under vanishing higher viscosity $\mu _1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p > \frac{3n}{n+2}$, its uniqueness and regularity for $p \ge 1 + \frac{2n}{n+2}$. In the case of the second model the existence of the weak solution is proved for $p>1$.
References:
[1] R. A. Adams: Sobolev Spaces. Academic Press, 1975. MR 0450957 | Zbl 0314.46030
[2] H. Bellout, F. Bloom, J. Nečas: Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids. Preprint, 1991. MR 1301173
[3] H. Bellout, F. Bloom, J. Nečas: Phenomenological Behaviour of Multipolar Viscous Fluids. Quaterly of Applied Mathematics 54 (1992), no. 3, 559–584. MR 1178435
[4] M. E. Bogovskij: Solutions of Some Problems of Vector Analysis with the Operators div and grad. Trudy Sem. S. L. Soboleva (1980), 5–41. (Russian)
[5] N. Dunford, J. T. Schwarz: Linear Operators: Part I. General Theory. Interscience Publishers Inc., New York, 1958.
[6] H. Gajewski, K. Gröger, K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie Verlag, Berlin, 1974. MR 0636412
[7] J. Kurzweil: Ordinary Differential Equations. Elsevier, 1986. MR 0929466 | Zbl 0667.34002
[8] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Flow. Gordon and Beach, New York, 1969. MR 0254401
[9] D. Leigh: Nonlinear Continuum Mechanics. McGraw-Hill, New York, 1968.
[10] J. L. Lions: Qeulques méthodes de résolution des problèms aux limites non lineaires. Dunod, Paris, 1969. MR 0259693
[11] J. Málek, J. Nečas, A. Novotný: Measure-valued solutions and asymptotic behavior of a multipolar model of a boundary layer. Czech. Math. Journal 42 (1992), no. 3, 549–575. MR 1179317
[12] J. Málek, J. Nečas, M. Růžička: On Non-Newtonian Incompressible Fluids. M3AS 1 (1993). MR 1203271
[13] J. Nečas: An Introduction to Nonlinear Elliptic Equations. J. Wiley, 1984.
[14] J. Nečas: Theory of Multipolar Viscous Fluids. The mathematics of finite elements and applications VII, MAFELAP 1990, J. R. Whiteman (ed.), Academic Press, 1991, pp. 233–244. MR 1132501
[15] J. Nečas, M. Šilhavý: Multipolar Viscous Fluids. Quaterly of Applied Mathematics 49 (1991), no. 2, 247–266. MR 1106391
[16] M. Pokorný: Cauchy Problem for the Non-Newtonian Incompressible Fluid (Master degree thesis, Faculty of Mathematics and Physics, Charles University, Prague. 1993.
[17] K. R. Rajagopal: Mechanics of Non-Newtonian Fluids. G. P. Galdi, J. Nečas: Recent Developments in Theoretical Fluid Dynamics, Pitman Research Notes in Math. Series 291, 1993. MR 1268237 | Zbl 0818.76003
[18] R. Temam: Navier-Stokes Equations—Theory and Numerical Analysis. North Holland, Amsterodam-New York-Oxford, 1979. MR 0603444 | Zbl 0426.35003
[19] H. Triebel: Theory of Function Spaces. Birkhäuser Verlag, Leipzig, 1983. MR 0781540 | Zbl 0546.46028
[20] H. Triebel: Interpolation Theory, Function Spaces, Differential Operators. Verlag der Wiss., Berlin, 1978. MR 0500580 | Zbl 0387.46033

Partner of