Giovanni P. Galdi

An Introduction to the Navier-Stokes Initial-Boundary Value Problem

The equations of motion of an incompressible, Newtonian fluid — usually called Navier-Stokes equations — have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C.M.L.H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes 1845 in a quite general way, by means of the theory of continua.

A note on the existence and uniqueness of solutions of the micropolar fluid equations

Abstract We show that existence and uniqueness theorems, known in the theory of the Navier-Stokes equations, are valid for the incompressible micropolar equations too.

Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations

In this paper we consider the Cauchy problem for the n-dimensional Navier-Stokes equations and we prove a regularity criterion for weak solutions involving the summability of the pressure. Related results for the initial-boundary value problem are also presented.

APPROXIMATION OF THE LARGER EDDIES IN FLUID MOTIONS II: A MODEL FOR SPACE-FILTERED FLOW

We present two modifications of continuum models used in large eddy simulation. The first modification is a closure approximation which better attenuates small eddies. The second modification is a change in the boundary conditions for the large eddy model from strict adherence to slip with resistance. For model consistency, the resistance coefficient is a function of the averaging radius so that the models boundary conditions reduce to a no-slip as the averaging radius decreases to zero.

Further existence results for classical solutions of the equations of a second-grade fluid

Over the last twenty years, there has been a remarkable interest in the study of a class of non-Newtonian fluids, known as fluids of grade n, from both theoretical and experimental points of view, see, e.g., [7, 14, 15, 5, 9, 16, 19] and the references cited therein. For a general incompressible fluid of grade 2, the Cauchy stress T is given by T = --/~I + #A1 + ~1A2 + c~2A 2, (1.0) where ~t is the viscosity, ~ , c~2 are material coefficients (normal stress moduli), /~ represents the pressure and A~, A 2 a r e the first two Rivlin-Ericksen tensors [17, 20], defined by As = Vv + (Vv) r, d A2 = ~A1 + A1Vv + (vv)r A1.

Monotonic Decreasing and Asymptotic Behavior of the Kinetic Energy for Weak Solutions of the Navier-Stokes Equations in Exterior Domains

Existence of weak solutions of the three-dimensional Navier-Stokes problem was first proved by J. Leray in the case of the Cauchy problem, cf. Leray (1934). As is well known, these solutions are important in that they are the only solutions which, so far, are known to exist for all times, without restriction on the data. Unfortunately, however, the question of whether they are classical, in an ordinary sense, is still open, even though partial conclusions regarding regularity are avilable: Leray (1934), Scheffer (1976, 1980), Caffarelli, Kohn, & Nirenberg (1982). Subsequently, E. Hopf (1951), using a different technique, constructed weak solutions for a general initial-boundary value problem. However, hopf’s solution (even for the Cauchy problem) has weaker properties than Leray’s solution. Among others, we refer to the “energy inequality” for the velocity field u(x, t), i.e.

Weighted Energy Methods in Fluid Dynamics and Elasticity

ntimits_mega {{u^2}} (x,t)dx - ntimits_mega {{u^2}(x,s)sx} eqq - 2ntimits_s^t {ntimits_mega {abla u:abla udxdau } }

Mathematical Problems in Classical and Non-Newtonian Fluid Mechanics

(I) for s = 0, for almost all s > 0 and for all t ≧ s.