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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.
This volume brings together five contributions to mathematical
fluid mechanics, a classical but still very active research field
which overlaps with physics and engineering. The contributions
cover not only the classical Navier-Stokes equations for an
incompressible Newtonian fluid, but also generalized Newtonian
fluids, fluids interacting with particles and with solids, and
stochastic models. The questions addressed in the lectures range
from the basic problems of existence of weak and more regular
solutions, the local regularity theory and analysis of potential
singularities, qualitative and quantitative results about the
behavior in special cases, asymptotic behavior, statistical
properties and ergodicity.
Vsevolod Alekseevich Solonnikov is known as one of the outstanding
mathema-
ciansfromtheSt.PetersburgMathematicalSchool.Hisremarkableresultsonexact
estimates of solutions to boundary and initial-boundary value
problems for linear elliptic, parabolic, and Stokes systems, his
methods and contributions to the - vestigation of free boundary
problems, in particular in ?uid mechanics, are well known to
specialists all over the world. The International Conference on
"Trends in Partial Di?erential Equations of th ' Mathematical
Physics" was held on the occasion of his 70 birthday in Obidos
(Portugal), from June 7 to 10, 2003. It was an organization of the
"Centro de Matem' atica e Aplica, c" oes Fundamentais da
Universidade Lisboa", in collaboration with the "Centro de Matem'
atica da Universidade de Coimbra", the "Centro de Matem' atica
Aplicada do IST/Universidade T' ecnica de Lisboa", the "Centro de
Matem' atica da Universidade da Beira Interior",from Portugal,and
with the L- oratory of Mathematical Physics of the St.Petersburg
Department of the Steklov Institute of Mathematics from Russia. The
conference consisted of thirty eight invited and contributed
lectures and '
gathered,inthecharminganduniquemedievaltownofObidos,aboutsixtypart-
ipants from ?fteen countries, namely USA, Switzerland, Spain,
Russia, Portugal, Poland, Lithuania, Korea, Japan, Italy, Germany,
France, Canada, Australia and
Argentina.Severalcolleaguesgaveusahelpinghandintheorganizationofthec-
ference. We are thankful to all of them, and in particular to
Stanislav Antontsev, Anvarbek Meirmanov and Ad' elia Sequeira, that
integrated also the Organizing Committee. A special acknowledgement
is due to Elena Frolova that helped us in compiling the short and
necessarily incomplete bio-bibliographical notes below.
Variational methods are applied to prove the existence of weak
solutions for boundary value problems from the deformation theory
of plasticity as well as for the slow, steady state flow of
generalized Newtonian fluids including the Bingham and
Prandtl-Eyring model. For perfect plasticity the role of the stress
tensor is emphasized by studying the dual variational problem in
appropriate function spaces. The main results describe the analytic
properties of weak solutions, e.g. differentiability of velocity
fields and continuity of stresses. The monograph addresses
researchers and graduate students interested in applications of
variational and PDE methods in the mechanics of solids and fluids.
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