Abstract
We consider in this paper the equations of motion of third grade fluids on a bounded domain of \(\mathbb{R}^2\) or \(\mathbb{R}^3\) with Navier boundary conditions. Under the assumption that the initial data belong to the Sobolev space H 2, we prove the existence of a global weak solution. In dimension two, the uniqueness of such solutions is proven. Additional regularity of bidimensional initial data is shown to imply the same additional regularity for the solution. No smallness condition on the data is assumed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams R. (1975). Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York
Amrouche C., Cioranescu D. (1997). On a class of fluids of grade 3. Int. J. Non-Linear Mech. 32(1): 73–88
Bresch D., Lemoine J. (1999). On the existence of solutions for non-stationary third-grade fluids. Int. J. Non-Linear Mech. 34(1): 485–498
Busuioc V. (2002). The regularity of bidimensional solutions of the third grade fluids equations. Math. Comput. Modelling 35(7–8): 733–742
Busuioc V., Iftimie D. (2004). Global existence and uniqueness of solutions for the equations of third grade fluids. Int. J. Non-Linear Mech. 39(1): 1–12
Busuioc V., Ratiu T.S. (2003). The second grade fluid and averaged Euler equations with Navier-slip boundary conditions. Nonlinearity 16(3): 1119–1149
Camassa R., Holm D.D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11): 1661–1664
Chen S., Foias C., Holm D.D., Olson E., Titi E.S., Wynne S. (1998). The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81, 5338–5341
Dunn J.E., Fosdick R.L. (1974). Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Rational Mech. Anal. 56, 191–252
Fosdick R.L., Rajagopal K.R. (1980). Thermodynamics and stability of fluids of third grade. Proc. R. Soc. Lond., Ser. A 369, 351–377
Holm D.D., Marsden J.E., Ratiu T.S. (1998). The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137(1): 1–81
Holm D.D., Marsden J.E., Ratiu T.S. (1998). Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349, 4173–4177
Jäger W., Mikelić A. (2003). Couette flows over a rough boundary and drag reduction. Comm. Math. Phys. 232(3): 429–455
Jäger W., Mikelić A. (2001). On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Diff. Equat. 170(1): 96–122
Navier, C. L. M. H. (1827). Sur les lois de l’équilibre et du mouvement des corps élastiques. Mem. Acad. R. Sci. Inst. France 369.
Rivlin R.S., Ericksen J.L. (1955). Stress-deformation relations for isotropic materials. J. Rational Mech. Anal. 4, 323–425
Sequeira A., Videman J. (1995). Global existence of classical solutions for the equations of third grade fluids. J. Math. Phys. Sci. 29(2): 47–69
Serrin J. (1959). Mathematical Principles of Classical Fluid Mechanics. Handbuch der Physics, Vol. 8/1 Springer, Berlin, pp. 125–263
Solonnikov, V. A. (1977). The solvability of the second initial-boundary value problem for a linear nonstationary system of Navier-Stokes equations. (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions, 10. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 69, 200–218, 277.
Solonnikov, V. A. (1988). On the transient motion of an isolated volume of viscous incompressible fluid. Math. USSR, Izv. 31(2), 381–405; (1987). translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51(5), 1065–1087.
Solonnikov, V. A., and Scadilov, V. E. (1973). On a boundary value problem for a stationary system of Navier-Stokes equations. Proc. Steklov Inst. Math. 125, 186–199; Translation from Trudy Mat. Inst. Steklov 125, 196–210.
Temam R. (1983). Problèmes mathématiques en plasticité. Gauthier-Villars, Paris
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Busuioc, A.V., Iftimie, D. A Non-Newtonian Fluid with Navier Boundary Conditions. J Dyn Diff Equat 18, 357–379 (2006). https://doi.org/10.1007/s10884-006-9008-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-006-9008-3