Analytical properties of flows of second-gradient fluids

Navier-Stokes equations are since many years the most important tool for studying viscous fluids. They are quite well established under a physical point of view, providing anyway one of the most challenging problems in Analysis. During the last century a number of variants of the Navier-Stokes equations have been proposed, mainly with the goal of describing some nonlinear phenomena like, e.g. shear thinning, and they got a considerable success in describing features of some biological fluids, like blood. Yet, all those fluids, including of course Navier-Stokes, share one common property: the work expended by the inner forces depends only on first derivatives of the velocity, as it has to be, at least for simple fluids. However, starting first from a very general point of view with the pioneering work of Germain, who introduced in a systematic way the concept of virtual power and its use in the foundations of Continuum Mechanics, and subsequently with many others, it became clearer and clearer that another possibile generalization was available, i.e. the second-gradient fluids. In these fluids the working done by the inner forces depends also on the second derivatives of the velocity field and includes the possibility of a “hyperviscosity” analogous to hyperstress coefficients which appears in the corresponding solid mechanics theories. These fluids have been considered only as an exercise, or an analytical variation of the problem, until the works of Fried and Gurtin , Giusteri and Fried and Giusteri-Marzocchi-Musesti, in which some convincing features of physical materials of this type were described and used. Subsequently, the very important case of isotropic fluids showed that this generalization leads to what we will call Hyperviscous Navier-Stokes problem. In this Thesis we will deal with the general initial and boundary value problem of such a fluid in a bounded or unbounded domain in three spatial dimensions, which is still open (even for bounded domains) for Navier-Stokes, with natural homogeneous boundary conditions. We introduce the second-order fluids trough their derivation from the theory of Virtual Powers: the advantages of considering such fluids are presented, for example the possibility to treat slender bodies moving in viscous fluids. Then, we study the initial boundary value problem for the hyperviscous Navier-Stokes system, that describes the special class of fluids derived in such a theory; in particular, we will consider the flow of these fluids at first in bounded domains and then in exterior domains. In order to do this, the existence of a solution is proved trough the construction of a suitable Galerkin approximated solution that passes to the limit thanks to suitable a priori energy estimates which are independent on the size of the bounded domains, thus allowing also existence in unbounded domains. In these estimates, the terms which do not appear in the Navier-Stokes problem will play a crucial role. The solutions are then proved to be regular, both in time and space, and unique in their functional spaces.