218 References and remarks on rotating fluids
R. Strichartz, [13] and [14] by P. Brenner and [102] by H. Pecher. The reader
who wants to become more familiar with Strichartz estimates can refer to [66]
by J. Ginibre and G. Velo and [80] by M. Keel and T. Tao.
A huge literature exists concerning applications to non-linear problems. For
the Schr¨odinger equations, the literature is numerous. The book [26] provides a
nice introduction to the subject. For a recent example of such applications, we
refer for instance to [36].
Concerning the non-linear wave equation, the reader can refer to [104] for
semilinear equations, and [8] and [83] for quasilinear equations. Let us mention
the approach of commuting vector fields developed in [81] which does not require
us to write a parametrix. This type of inequality has been used in the context
of the incompressible limit for viscous fluids by B. Desjardins and E. Grenier
(see [47]) to prove the analog of Theorem 5.6, page 104. In [41], R. Danchin
proved the analog of Theorem 5.7 for the incompressible limit.
In the context of rotating fluids, the use of these techniques comes from [32]
where a weaker version of Theorems 5.6, page 104, and 5.7, page 108, are proved.
In the case of periodic boundary conditions, the first result of the type of
Theorem 6.2, page 119, was proved in 1996 by A. Babin, A. Mahalov and B.
Nicolaenko in [5] under a non-resonance condition (namely condition (R) intro-
duced in Definition 6.2, page 144). Then the same authors dropped this condition
in 1999 (see [6]) and proved Theorem 7.2, page 157. Moreover, asymptotic expan-
sions in ε have been proved by I. Gallagher in 1998 (see [59] and [60]). Let us note
that in the context of the incompressible limit, there is no such non-resonance
condition. In spite of that, N. Masmoudi proved in [96] that the limit system
(which is surprisingly globally parabolic as proved by I. Gallagher in [61]) in that
case is globally well-posed. Using that, R. Danchin proved in [42] the analog of
Theorem 6.2, page 119.
For Ekman boundary layers, the pioneering mathematical work is the work by
N. Masmoudi and E. Grenier (see [72]) where Theorem 7.1, page 156, is proved.
This corresponds to the case of well-prepared data. The case of ill-prepared data
with horizontal periodic boundary conditions was investigated by N. Masmoudi
in [95]. Theorem 7.2, page 157, was proved by the authors in [34].