Global analysis for the fluids of a power-law type (Q2754442)

A critical survey is presented of results on the well-posedness of systems of partial differential equations governing the flows of incompressible fluids with shear-dependent viscosity. The focus is on global existence, uniqueness, and continuous dependence on the data. The governing equations are the usual equations of momentum and incompressibility, where in particular the stress \({\mathbf T}\) is given by \(\mathbf T = 2\mu {\mathbf D}\), in which \({\mathbf D}\) is the rate of deformation tensor, and \(\mu\) is the dynamic viscosity, which is assumed to be a function of \(|{\mathbf D}|^2\). NEWLINENEWLINENEWLINEThe standard results for the Navier-Stokes equations, that is, corresponding to constant viscosity, are summarized: in two dimensions one has global well-posedness; in three dimensions, the question is unresolved, with the main contribution being that of Leray 70 years ago, in which existence is established for certain weak solutions. NEWLINENEWLINENEWLINEA typical functional form for \(\mu\) is that of \(\mu = \mu_0 (1 + \varepsilon |{\mathbf D}|^{p-2})\). Three cases are then relevant: (a) when \(p=2\), one has the Navier Stokes equations; (b) for \(p \geq 2\) one would expect good global properties; and (c) for \(p < 2\), it would be a nontrivial matter to make significant progress. NEWLINENEWLINENEWLINEThree kinds of domains \(\Omega\) are considered: bounded domains; domains such that the problem is space-periodic; and the Cauchy problem, for which \(\Omega = \mathbb{R}^d\). NEWLINENEWLINENEWLINEBy combining the theory of monotone operators with compactness arguments, one obtains an existence result, due to Ladyzhenskaya, on the existence of weak solutions when \(p \geq (3d +2)/(d +2)\), and uniqueness when \(p \geq (d + 2)/2\), for the Dirichlet problem on a bounded domain. Thus the bounds coincide for \(d = 2\). NEWLINENEWLINENEWLINEThe author discusses three further methods, which have been introduced with a view to extend the results due to Ladyzhenskaya. First, in the regularity method, the development of certain a priori estimates leads to the existence of a weak solution, for the Cauchy and periodic problems, for \(p \geq 3d/(d + 2)\). Furthermore, one obtains the existence of a regular (strong) solution for \(p \geq (3d + 2)/(d + 2)\). NEWLINENEWLINENEWLINEThe truncation method is based on the construction of a special test function, and improved results are obtained for weak solutions, for the steady Dirichlet problem, for \(p > 2d/(d+2)\). NEWLINENEWLINENEWLINEThe use of the regularity method for the unsteady Dirichlet problem leads to results on the existence of weak solutions for \(d = 3\), and uniqueness and smoothness when \(p > 9/4\). The question of Hölder continuity of the velocity gradient is considered in detail, and results are summarized for the two-dimensional problem. NEWLINENEWLINENEWLINEA short section is devoted to the question of the nature of attractors. Using the method of \(\ell\)-trajectories, it can be shown that for \(p \geq (3d + 2)/(d + 2)\), the periodic problem possesses a global attractor with finite fractal dimension, and an exponential attractor. NEWLINENEWLINENEWLINEThe paper concludes with a review of results for the Navier-Stokes equations that are related to the results in earlier sections, on the generalized problem. NEWLINENEWLINENEWLINEThe author gives a comprehensive citation of relevant works in the literature, but unfortunately, references [32], [55] are missing from the bibliography.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00018].