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A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. (English) Zbl 1167.76316
Summary: We consider a complete thermodynamic model for unsteady flows of incompressible homogeneous Newtonian fluids in a fixed bounded three-dimensional domain. The model comprises evolutionary equations for the velocity, pressure and temperature fields that satisfy the balance of linear momentum and the balance of energy on any (measurable) subset of the domain, and is completed by the incompressibility constraint. Finding a solution in such a framework is tantamount to looking for a weak solution to the relevant equations of continuum physics. If in addition the entropy inequality is required to hold on any subset of the domain, the solution that fulfills all these requirements is called the suitable weak solution. In our setting, both the viscosity and the coefficient of the thermal conductivity are functions of the temperature. We deal with Navier’s slip boundary conditions for the velocity that yield a globally integrable pressure, and we consider zero heat flux across the boundary. For such a problem, we establish the large-data and long-time existence of weak as well as suitable weak solutions, extending thus J. Leray’s [Acta Math. 63, 193–248 (1934; JFM 60.0726.05)] and L. Caffarelli, R. Kohn and L. Nirenberg’s [Commun. Pure Appl. Math. 35, 771–831 (1982; Zbl 0509.35067)] results, that deal with the problem in a purely mechanical context, to the problem formulated in a fully thermodynamic setting.

MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q35 PDEs in connection with fluid mechanics
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References:
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