# Cauchy problem for the non-newtonian viscous incompressible fluid

Applications of Mathematics (1996)

- Volume: 41, Issue: 3, page 169-201
- ISSN: 0862-7940

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topPokorný, Milan. "Cauchy problem for the non-newtonian viscous incompressible fluid." Applications of Mathematics 41.3 (1996): 169-201. <http://eudml.org/doc/32944>.

@article{Pokorný1996,

abstract = {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 $p>1$.},

author = {Pokorný, Milan},

journal = {Applications of Mathematics},

keywords = {non-Newtonian incompressible fluids; Navier-Stokes equations; Cauchy problem; non-Newtonian fluids; bipolar fluids; existence, uniqueness, regularity of weak solution},

language = {eng},

number = {3},

pages = {169-201},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Cauchy problem for the non-newtonian viscous incompressible fluid},

url = {http://eudml.org/doc/32944},

volume = {41},

year = {1996},

}

TY - JOUR

AU - Pokorný, Milan

TI - Cauchy problem for the non-newtonian viscous incompressible fluid

JO - Applications of Mathematics

PY - 1996

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 41

IS - 3

SP - 169

EP - 201

AB - 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 $p>1$.

LA - eng

KW - non-Newtonian incompressible fluids; Navier-Stokes equations; Cauchy problem; non-Newtonian fluids; bipolar fluids; existence, uniqueness, regularity of weak solution

UR - http://eudml.org/doc/32944

ER -

## References

top- Sobolev Spaces, Academic Press, 1975. (1975) Zbl0314.46030MR0450957
- Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids, Preprint, 1991. (1991) MR1301173
- 10.1090/qam/1178435, Quaterly of Applied Mathematics 54 (1992), no. 3, 559–584. (1992) MR1178435DOI10.1090/qam/1178435
- Solutions of Some Problems of Vector Analysis with the Operators div and grad, Trudy Sem. S. L. Soboleva (1980), 5–41. (Russian) (1980)
- Linear Operators: Part I. General Theory, Interscience Publishers Inc., New York, 1958. (1958)
- Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie Verlag, Berlin, 1974. (1974) MR0636412
- Ordinary Differential Equations, Elsevier, 1986. (1986) Zbl0667.34002MR0929466
- The Mathematical Theory of Viscous Flow, Gordon and Beach, New York, 1969. (1969) MR0254401
- Nonlinear Continuum Mechanics, McGraw-Hill, New York, 1968. (1968)
- Qeulques méthodes de résolution des problèms aux limites non lineaires, Dunod, Paris, 1969. (1969) MR0259693
- Measure-valued solutions and asymptotic behavior of a multipolar model of a boundary layer, Czech. Math. Journal 42 (1992), no. 3, 549–575. (1992) MR1179317
- On Non-Newtonian Incompressible Fluids, M3AS 1 (1993). (1993) MR1203271
- An Introduction to Nonlinear Elliptic Equations, J. Wiley, 1984. (1984)
- 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. (1991) MR1132501
- 10.1090/qam/1106391, Quaterly of Applied Mathematics 49 (1991), no. 2, 247–266. (1991) MR1106391DOI10.1090/qam/1106391
- Cauchy Problem for the Non-Newtonian Incompressible Fluid (Master degree thesis, Faculty of Mathematics and Physics, Charles University, Prague, 1993. (1993)
- Mechanics of Non-Newtonian Fluids, G. P. Galdi, J. Nečas: Recent Developments in Theoretical Fluid Dynamics, Pitman Research Notes in Math. Series 291, 1993. (1993) Zbl0818.76003MR1268237
- Navier-Stokes Equations—Theory and Numerical Analysis, North Holland, Amsterodam-New York-Oxford, 1979. (1979) Zbl0426.35003MR0603444
- Theory of Function Spaces, Birkhäuser Verlag, Leipzig, 1983. (1983) Zbl0546.46028MR0781540
- Interpolation Theory, Function Spaces, Differential Operators, Verlag der Wiss., Berlin, 1978. (1978) Zbl0387.46033MR0500580

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