Miroslav Bulíček, Roger Lewandowski, Josef Málek
On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions

Comment.Math.Univ.Carolin. 52,1 (2011) 89-114.

Abstract:In this paper, we establish the large-data and long-time existence of a suitable weak solution to an initial and boundary value problem driven by a system of partial differential equations consisting of the Navier-Stokes equations with the viscosity $\nu $ polynomially increasing with a scalar quantity $k$ that evolves according to an evolutionary convection diffusion equation with the right hand side $\nu (k)|{\pmb{\mathsf{D}}}(\vec{v})|^2$ that is merely $L^1$-integrable over space and time. We also formulate a conjecture concerning regularity of such a~solution.

Keywords: large data existence, suitable weak solution, Navier-Stokes-Fourier equations, incompressible fluid, the viscosity increasing with a scalar quantity, regularity, turbulent kinetic energy model
AMS Subject Classification: 35Q30 35Q35 76F60