Geodesic
On the Navier–Stokes equations with the energy-dependent nonlocal viscosities
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 71-91 
L. Consiglieri; J.-F. Rodrigues; T. N. Shilkin. On the Navier–Stokes equations with the energy-dependent nonlocal viscosities. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 34, Tome 306 (2003), pp. 71-91. http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a3/
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     title = {On the {Navier{\textendash}Stokes} equations with the energy-dependent nonlocal viscosities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     year = {2003},
     volume = {306},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_306_a3/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We discuss the mathematical derivation of incompressible viscous flows where the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we consider first the case of temperature dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain as an asymptotic limit the Navier–Stokes system coupled with and ordinary differential equation involving the dissipation energy. Letting further the latent heat vanish, we derive the Navier–Stokes equations with a nonlocal viscosity depending on the total dissipation of energy.

[1] L. Consiglieri, J. F. Rodrigues, “On stationary flows with energy dependent nonlocal viscosities”, Zap. Nauchn. Semin. POMI, 295, 2003, 99–117 | MR | Zbl

[2] D. Hilhorst, J. F. Rodrigues, “On a nonlocal diffusion equation with discontinuous reaction”, Adv. Diff. Eqs., 5:4–6 (2000), 657–680 | MR | Zbl

[3] O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of a Viscous Incompressible Fluid, 2nd, Nauka, Moscow, 1970

[4] O. A. Ladyzhenskaya, “On new equations for the description of the motion of viscous incompressible fluids and on the solvability “in large” of some boundary-value problems for them”, Proceedings of Steklov Math. Inst., 102, 1967, 85–104 | MR | Zbl

[5] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, Amer. Math. Soc., Providence, RI, 1967 | MR

[6] J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Nonlineaires, Gauthier-Villars, Paris, 1969 | MR | Zbl

[7] J. Simon, “Compact sets in the space $L^p(0,T; B)$”, Ann. Mat. Pura Appl., 146:4 (1987), 65–96 | MR | Zbl