4
References and remarks on the Navier–Stokes equations
The purpose of this chapter is to give some historical landmarks to the reader.
The concept of weak solutions certainly has its origin in mechanics; the article
by C. Oseen [100] is referred to in the seminal paper [87] by J. Leray. In that
famous article, J. Leray proved the global existence of solutions of (NS
ν
)inthe
sense of Definition 2.5, page 42, in the case when Ω = R
3
. The case when Ω is
a bounded domain was studied by E. Hopf in [74]. The study of the regularity
properties of those weak solutions has been the purpose of a number of works.
Among them, we recommend to the reader the fundamental paper of L. Caffarelli,
R. Kohn and L. Nirenberg [21]. In two space dimensions, J.-L. Lions and G. Prodi
proved in [91] the uniqueness of weak solutions (this corresponds to Theorem 3.2,
page 56, of this book). Theorem 3.3, page 58, of this book shows that regularity
and uniqueness are two closely related issues. In the case of the whole space R
3
,
theorems of that type have been proved by J. Leray in [87]. For generalizations
of that theorem we refer to [113], [121], [58] and [62]. In the article [87], J. Leray
also proved the global regularity of weak solutions (and their global uniqueness)
for small initial data, namely initial data satisfying
u
0
2
L
2
u
0
L
∞
≤ c
3
ν
3
or u
0
2
L
2
∇u
0
L
2
≤ c
2
ν
2
.
Theorem 3.4, page 66, and Theorem 3.5, page 73, which are generalizations
of the first smallness condition above, were proved by H. Fujita and T. Kato
in [57]. In the case if the whole space R
3
, the smallness condition has been
generalized in terms of the Lebesgue space L
3
by T. Kato in [79], of Besov
spaces by M. Cannone, Y. Meyer and F. Planchon in [25], and of the BMO-type
space BMO
−1
by H. Koch and D. Tataru in [84]. The set of results presented in
this part contains the material required for the further study of rotating fluids.
The reader who wants to learn more about the theory of the incompressible
Navier–Stokes system can read the following monographs:
• M. Cannone: Ondelettes, paraproduits et Navier–Stokes [24]
• J.-Y. Chemin: Localization in Fourier space and Navier–Stokes system [30]
• P. Constantin and C. Foias: Navier–Stokes Equations [38]
• P.-G. Lemari´e-Rieusset: Recent Developments in the Navier–Stokes
problem [86]
• P.-L. Lions: Mathematical topics in fluid mechanics [92]