Weak Solutions, A Priori Estimates

Abstract

The fundamental laws of continuum mechanics interpreted as infinite families of integral identities introduced in Chapter 1, rather than systems of partial differential equations, give rise to the concept of weak (or variational) solutions that can be vastly extended to extremely divers physical systems of various sorts. The main stumbling block of this approach when applied to the field equations of fluid mechanics is the fact that the available a priori estimates are not strong enough in order to control the flux of the total energy and/or the dissipation rate of the kinetic energy. This difficulty has been known since the seminal work of Leray [132] on the incompressible Navier-Stokes system, where the validity of the so-called energy equality remains an open problem, even in the class of suitable weak solutions introduced by Caffarelli et al. [37]. The question is whether or not the rate of decay of the kinetic energy equals the dissipation rate due to viscosity as predicted by formula (1.39). It seems worth noting that certain weak solutions to hyperbolic conservation laws indeed dissipate the kinetic energy whereas classical solutions of the same problem, provided they exist, do not. On the other hand, however, we are still very far from complete understanding of possible singularities, if any, that may be developed by solutions to dissipative systems studied in fluid mechanics. The problem seems even more complex in the framework of compressible fluids, where Hoff [113] showed that singularities survive in the course of evolution provided they were present in the initial data. However, it is still not known if the density may develop “blow up” (gravitational collapse) or vanish (vacuum state) in a finite time. Quite recently, Brenner [28] proposed a daring new approach to fluid mechanics, where at least some of the above mentioned difficulties are likely to be eliminated.