On the Navier-Stokes Equations with Energy-Dependent Nonlocal Viscosities

We discuss the mathematical modeling of incompressible viscous flows for which the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we begin with the case of temperature-dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain the Navier-Stokes system coupled with an ordinary differential equation involving the dissipation energy as the asymptotic limit. Letting further the latent heat to vanish, we derive the Navier-Stokes equations with a nonlocal viscosity depending on the total dissipation of energy. Bibliography: 7 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 71–91.