Some Aspects of a Mathematically Rigorous Theory
In this chapter the reader can find some basic tools for the rigorous analysis of the viscous (mainly Navier) equations.
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In this chapter the reader can find some basic tools for the rigorous analysis of the viscous (mainly Navier) equations. In particular, for definiteness and simplicity, we consider the Galerkin approximation and weak solutions for the Navier equations, with periodic boundary conditions on {l = [0, LJd, in the presence of a body force which is constant in time. Unfortunately, the weak solutions of the Navier equations are not unique in any sense! Their construction as the limit of a subsequence of the Galerkin approximation leaves open the possibility that there is more than one distinct limit, even for the same sequence of approximations. Nonunique evolution would violate the basic tenets of classical Newtonian determinism and would render the Navier equations worthless as a predictive model. Naturally, if the weak solutions were smooth enough that all of the terms in the Navier equations made sense as "normal" functions, then we would say that the weak solutions were "strang" solutions. As noted pertinently by Doering and Gibbon (1995), "To many readers this kind of distinction may seem little more than a mathemaical formality of no real consequence or practical importance. The issues involved, however, go straight to the heart of the question of the validity and self-consistency of the Navier equations as a hydrodynamical model, and the mathematical difficulties have their source in precisely the same physical phenomena that the equations are meant to describe"! In this chapter we also give some information concerning recent rigorous mathematical results (existence, regularity, and uniqueness) for incompressible and compressible viscous fluid flow problems. In the two recent review papers by Zeytounian (1999, 2001), the reader can find a fluid-dynamic point of view on some mathematical aspects of fluid flows. For the mathematically rigorous theory of the Navier incompressible viscous system, we mention the books by Ladhyzhenskaya (1969), Shinbrot (1973), Temam (1984), Constantin and Foias (1988), Galdi (1994/1998a,b), Doering and Gibbon (1995), P.-L. Lions (1996), and the recent survey by Temam (2000) which describes the Navier equations from the work of Leray in 1933 up to the recent research on attractors and turbulence. It is also necesR. K. Zeytounian, Theory and Applications of Viscous Fluid Flows © Springer-Verlag Berlin Heidelberg 2004
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sary to mention the recent, very pertinent, short paper of Jean-Yves Chemin (2000), where the author gives an illuminating analysis of the Leray (1934) fundamental paper: "Sur le mouvement d'un liquide visqueux emplissant l'espace", published in Acta Math., 63 (1934), 193-248, and also the survey papers by Heywood (1976, 1980a,b,c, 1989). Concerning the mathematical legacy of Jean Leray see also Temam (1999). For mathematically rigorous results of compressible viscous equations, we mention the book of P.-L. Lions (1998) and the review papers by Solonnikov and Kazhikov (1981) and Valli (1992). In the recent book by Le
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