Geodesic
On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 89-114.

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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.
Classification : 35Q30, 35Q35, 76F60
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 
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Bulíček, Miroslav; Lewandowski, Roger; Málek, Josef. On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 1, pp. 89-114. http://geodesic.mathdoc.fr/item/CMUC_2011__52_1_a6/