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Mathematical issues concerning the Navier-Stokes equations and some of its generalizations. (English) Zbl 1095.35027
Dafermos, C.M.(ed.) et al., Evolutionary equations. Vol. II. Amsterdam: Elsevier/North-Holland (ISBN 0-444-52048-1/hbk). Handbook of Differential Equations, 371-459 (2005).
This paper deals with unsteady and incompressible flows of viscous fluids with constant or pressure and shear dependent viscosity. The movements are considered from a more general point of view than that of viscous fluids, so that the Navier-Stokes fluids are particular cases.
The paper is divided into two parts. In the first part there are presented incompressible fluids with shear, pressure and density dependent viscosity from the point of view of continuum mechanics and, in the second part, the mathematical analysis of these fluid flows is made. Concerning the first part, it is sufficient to say that it begins with the subsection “What is a fluid?” and ends with the section “Boundary conditions”, to underline the mode in which the authors conduct their analysis. They begin with Newton’s considerations (1687), pass through those of Du Buat (1779) and continue with G. D. Poisson’s (1831), Saint-Venant’s (1843), F. T. Tronton’s (1906) and end with their own ones. They present a Navier-Stokes fluid model, start from Newtonian behaviour, balance of mass, incompressibility, inhomogeneity, linear moment, angular momentum, energy and continue with thermodynamic considerations and constitutive models for compressible and incompressible Navier-Stokes fluids.
In the second part the authors begin with a valuation of the models, define the mathematical self-consistency of the models, weak solutions and study the problem $$({\mathcal P})$$ and the approximate problem $$({\mathcal P}^{\varepsilon,\eta})$$. A great attention is given to the spaces in which the solutions are searched and to the special equations of viscous flows, especially Ladyzhenskaya’s equations. The paper ends with general considerations about other incompressible fluid models with pressure dependent or pressure and share dependent viscosities and with inhomogeneous incompressible fluids.
For the entire collection see [Zbl 1074.35003].

##### MSC:
 35Q30 Navier-Stokes equations 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics 35B41 Attractors 46N20 Applications of functional analysis to differential and integral equations