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We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called $L^{\infty }$-truncation method, used to obtain the strong convergence of the velocity...

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 $\nu \left(k\right)|𝖣\left(\overrightarrow{v}\right){|}^2$ that is merely $L^1$-integrable over space and time. We also formulate a conjecture concerning regularity...