||The imagination is stricken by the substantial conceptual identity between the problems met in the theoretical study of physical phenomena. It is absolutely unexpected and surprising, whether one studies equilibrium statistical mechanics, or quantum field theory, or solid state physics, or celestial mechanics, harmonic analysis, elasticity, general relativity or fluid mechanics and chaos in turbulence.
So that when in 19881 was made chair of Fluid Mechanics at the Universita La Sapienza, not to recognize work I did on the subject (there was none) but, rather, to avoid my teaching mechanics, from which I could have a strong cultural influence on mathematical physics in Roma, I was not excessively worried, although I was clearly in the wrong place. The subject is wide, hence in the last decade I could do nothing else but go through books and libraries looking for something that was within the range of the methods and experiences of my past work.
The first great surprise was to realize that the mathematical theory of fluids is in a state even more primitive than I was conscious of. Nevertheless it still seems to me that a detailed analysis of the mathematical problems is essential for any one who wishes research into fluids. Therefore I dedicated (Chap.3) all the space necessary to a complete exposition of the theories of Leray, of Scheffer and of Caffarelli, Kohn and Nirenberg, taken directly from the original works.
The analysis is preceded by a long discussion of the phenomenological aspects concerning the fluid equations and their properties, with particular attention to the meaning of the various approximations. One should not forget that the fluid equations do not have fundamental nature, i.e. they ultimately are phenomenological equations and for this reason one "cannot ask from them too much". In order to pose appropriate questions it is necessary to dominate the heuristic and phenomenological aspects of the theory. I could not do better than follow the Landau-Lifshitz volume, selecting from it a small, coherent set of properties without (obviously) being able nor wishing to reproduce it (which, in any event, would have been useless), leaving aside most of the themes covered by that rich, agile and modern treatise, which the reader will not set aside in his introductory studies.
In the introductory material (Chaps. 1,2) I inserted several modern remarks taken from works that I have come to know either from colleagues or from participating in conferences (or reading the literature). Here and there, rarely, there are a few original comments and ideas (in the sense that I did not find them in the accessed literature).
The second part of the book is dedicated to the qualitative and phenomenological theory of the incompressible Navies-Stokes equation: the lack of existence and uniqueness theorems (in three space dimensions) did not have