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The primary objective of this monograph is to develop an elementary
and se-
containedapproachtothemathematicaltheoryofaviscousincompressible?uid
n in a domain ? of the Euclidean spaceR, described by the equations
of Navier- Stokes. The book is mainly directed to students familiar
with basic functional analytic tools in Hilbert and Banach spaces.
However, for readers' convenience, in the ?rst two chapters we
collect, without proof some fundamental properties of Sobolev
spaces, distributions, operators, etc. Another important objective
is to formulate the theory for a completely general domain ?. In
particular, the theory applies to arbitrary unbounded, non-smooth
domains. For this reason, in the nonlinear case, we have to
restrict ourselves to space dimensions n=2,3 that are also most
signi?cant from the physical point of view. For mathematical
generality, we will develop the l- earized theory for all n? 2.
Although the functional-analytic approach developed here is, in
principle, known to specialists, its systematic treatment is not
available, and even the
diverseaspectsavailablearespreadoutintheliterature.However,
theliterature is very wide, and I did not even try to include a
full list of related papers, also because this could be confusing
for the student. In this regard, I would like to apologize for not
quoting all the works that, directly or indirectly, have inspired
this monograph.
The primary objective of this monograph is to develop an elementary
and se-
containedapproachtothemathematicaltheoryofaviscousincompressible?uid
n in a domain ? of the Euclidean spaceR , described by the
equations of Navier- Stokes. The book is mainly directed to
students familiar with basic functional analytic tools in Hilbert
and Banach spaces. However, for readers' convenience, in the ?rst
two chapters we collect, without proof some fundamental properties
of Sobolev spaces, distributions, operators, etc. Another important
objective is to formulate the theory for a completely general
domain ?. In particular, the theory applies to arbitrary unbounded,
non-smooth domains. For this reason, in the nonlinear case, we have
to restrict ourselves to space dimensions n=2,3 that are also most
signi?cant from the physical point of view. For mathematical
generality, we will develop the l- earized theory for all n? 2.
Although the functional-analytic approach developed here is, in
principle, known to specialists, its systematic treatment is not
available, and even the
diverseaspectsavailablearespreadoutintheliterature.However,theliterature
is very wide, and I did not even try to include a full list of
related papers, also because this could be confusing for the
student. In this regard, I would like to apologize for not quoting
all the works that, directly or indirectly, have inspired this
monograph.
This book offers an elementary, self-contained approach to the
mathematical theory of viscous, incompressible fluid in a domain of
the Euclidian space, described by the equations of Navier-Stokes.
It is the first to provide a systematic treatment of the subject.
It is designed for students familiar with basic tools in Hilbert
and Banach spaces, but fundamental properties of, for example,
Sobolev spaces, are collected in the first two chapters.
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