We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor ${\tau }^V\left(𝕖\right)=\tau \left(𝕖\right)-2{\mu }_1\Delta 𝕖$, where the nonlinear function $\tau \left(𝕖\right)$ satisfies ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge c{\left|𝕖\right|}^p$ or ${\tau }_{ij}\left(𝕖\right)e_{ij}\ge {c\left(\right|𝕖|}^2+{\left|𝕖\right|}^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p>1$ for both models. Then, under vanishing higher viscosity ${\mu }_1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p>\frac{3n}{n+2}$, its uniqueness and...

Continuum mechanics (e.g., hydrodynamics, elasticity theory) is based on the assumption that a small set of fields provides a closed description on large space and time scales. Conditions governing the choice for these fields are discussed in the context of granular fluids and multi-component fluids. In the first case, the relevance of temperature or energy as a hydrodynamic field is justified. For mixtures, the use of a total temperature and single...

We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...