Geodesic
On stationary flows with energy dependent nonlocal viscosities
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 99-117 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A nonlocal constitutive law for an incompressible viscous flow in which the viscosity depends on the total dissipation energy of the fluid is obtained as a limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated in an Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. The extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the framework of the heat equation with a $L^1$-term. The nonlocal model proposed by Ladyzenskaya in 1966 as a modification of Navier–Stokes equations, in particular, may be obtained with this procedure. 
@article{ZNSL_2003_295_a4,
     author = {L. Consiglieri and J.-F. Rodrigues},
     title = {On stationary flows with energy dependent nonlocal viscosities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {99--117},
     year = {2003},
     volume = {295},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/}
}
TY  - JOUR
AU  - L. Consiglieri
AU  - J.-F. Rodrigues
TI  - On stationary flows with energy dependent nonlocal viscosities
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2003
SP  - 99
EP  - 117
VL  - 295
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/
LA  - en
ID  - ZNSL_2003_295_a4
ER  - 
%0 Journal Article
%A L. Consiglieri
%A J.-F. Rodrigues
%T On stationary flows with energy dependent nonlocal viscosities
%J Zapiski Nauchnykh Seminarov POMI
%D 2003
%P 99-117
%V 295
%U http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/
%G en
%F ZNSL_2003_295_a4
L. Consiglieri; J.-F. Rodrigues. On stationary flows with energy dependent nonlocal viscosities. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 33, Tome 295 (2003), pp. 99-117. http://geodesic.mathdoc.fr/item/ZNSL_2003_295_a4/

[1] M. Chipot, J. F. Rodrigues, “On a class of Nonlocal Nonlinear Elliptic Problems”, RAIRO Modél. Math. Anal. Numér., 26 (1992), 447–468 | MR

[2] L. Consiglieri, “Stationary weak solutions for a class of non-Newtonian fluids with energy transfer”, Int. J. Non-Linear Mechanics, 32 (1997), 961–972 | DOI | MR | Zbl

[3] G. Duvaut, J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972 | MR | Zbl

[4] I. Ekeland, R. Temam, Analyse convexe et problèmes variationnels, Dunod et Gauthier-Villars, Paris, 1974 | MR | Zbl

[5] J. Frehse, J. Málek, M. Steinhauer, “An existence result for fluids with shear dependent viscosity-steady flows”, Nonlinear Analysis, Theory Meth. Appl., 30 (1997), 3041–3049 | DOI | MR | Zbl

[6] M. Fuchs, G. Seregin, Variational Methods from Problems of Plasticity Theory and for Generalized Newtonian Fluids, Lectures Notes in Mathematics, 1749, Springer, Berlin, 2000 | MR | Zbl

[7] Proc. Steklov Inst. Math., 102, 1967 | MR

[8] The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1969 | MR

[9] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969 | MR | Zbl

[10] J. Málek, J. Nečas, M. Rokyta, M. Ruužička, Weak and Measure-valued solutions to evolutionary PDEs, Chapman and Hall, London, 1996 | MR | Zbl

[11] J. Málek, K. R. Rajagopal, M. Ruužička, “Existence and regularity of solutions and stability of the rest state for fluids with shear dependent viscosity”, Math. Models Methods Appl. Sci., 6 (1995), 789–812 | DOI | MR | Zbl

[12] N. G. Meyers, “An $L^p$-estimates for the gradient of solutions of second order elliptic divergence equations”, Ann. Sci. Num. Sup. Pisa, 17 (1963), 189–206 | MR | Zbl

[13] A. Prignet, “Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure”, Ann. Fac. Sci. Toulouse Math., 6 (1997), 297–318 | MR | Zbl

[14] J. F. Rodrigues, “Reaction-Diffusion: from systems to nonlocal equations in a class of free boundary problems”, Reaction-Diffusion Systems: Theory and Applications, Proc. Int. Conf., RIMS, 1249, eds. M. Mimura, H. Okamoto, Univ. Kyoto, Japan, 2002 | MR