The lecture notes in this book are based on the TCC (Taught Course
Centre for graduates) course given by the author in Trinity Terms
of 2009-2011 at the Mathematical Institute of Oxford University. It
contains more or less an elementary introduction to the
mathematical theory of the Navier-Stokes equations as well as the
modern regularity theory for them. The latter is developed by means
of the classical PDE's theory in the style that is quite typical
for St Petersburg's mathematical school of the Navier-Stokes
equations.The global unique solvability (well-posedness) of initial
boundary value problems for the Navier-Stokes equations is in fact
one of the seven Millennium problems stated by the Clay
Mathematical Institute in 2000. It has not been solved yet.
However, a deep connection between regularity and well-posedness is
known and can be used to attack the above challenging problem. This
type of approach is not very well presented in the modern books on
the mathematical theory of the Navier-Stokes equations. Together
with introduction chapters, the lecture notes will be a
self-contained account on the topic from the very basic stuff to
the state-of-art in the field.
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