A Bound from Below on the Temperature for the Navier-Stokes-Fourier System

We give a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system, under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of [16], we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for $L^\infty$ bounds of solutions to elliptic equations with bounded measurable coefficients.

[1]  P. Lions.  Mathematical topics in fluid mechanics , 1996 .

[2]  Gui-Qiang G. Chen,et al.  Nonlinear Partial Differential Equations and Related Analysis , 2005 .

[3]  R. Kohn,et al.  Partial regularity of suitable weak solutions of the navier‐stokes equations , 1982 .

[4]  E. Feireisl,et al.  On a simple model of reacting compressible flows arising in astrophysics , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  E. Feireisl.  On the motion of a viscous, compressible, and heat conducting fluid , 2004 .

[6]  E. Feireisl,et al.  Singular Limits in Thermodynamics of Viscous Fluids , 2009 .

[7]  L. Caffarelli,et al.  The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics , 2010 .

[8]  C. Chan.  Smoothness Criterion for Navier-Stokes Equations in Terms of Regularity along the Streamlines , 2010 .

[9]  Eduard Feireisl,et al.  Mathematical theory of compressible, viscous, and heat conducting fluids , 2007, Comput. Math. Appl..

[10]  E. Feireisl,et al.  Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids , 2011 .

[11]  Eduard Feireisl,et al.  Dynamics of Viscous Compressible Fluids , 2004 .

[12]  Alexis Vasseur,et al.  A new proof of partial regularity of solutions to Navier-Stokes equations , 2007 .

[13]  E. Feireisl.  Stability of Flows of Real Monoatomic Gases , 2006 .

[14]  L. Caffarelli,et al.  Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation , 2006, math/0608447.

[15]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[16]  Pierre-Louis Lions,et al.  Mathematical Topics in Fluid Mechanics: Volume 2: Compressible Models , 1998 .