“ATTRACTORS
FOR HYDRODYNAMICAL PROBLEMS IN UNBOUNDED DOMAINS”

The concept of a
global attractor plays a very important role in the modern theory of dissipative
systems generated by PDEs. In particular, for the case of equations in bounded domains
(like a square, disk or ball), this attractor is usually finite-dimensional.
Thus, despite the infinite-dimensionality of the initial phase space, the limit dynamics reduced to
the attractor occurs finite-dimensional and can be studied by the methods of
the classical dynamical systems theory.

The situation is much
more difficult for the case where the underlying domain is unbounded (like R^2 or
R^3) since the dimension of the attractor is typically infinite here and
the finite-dimensional reduction
is no more possible. Nevertheless, a reasonable theory has been recently
developed for large class of such
systems. This theory includes sharp upper and
lower bounds for the Kolmogorov's
entropy of infinite-dimensional attractors, usage of essentially unstable manifolds
for non-hyperbolic equilibria, description of space
and space-time chaos via the Bernoulli schemes with infinite number of symbols, etc.

The aim of the project is to extend/apply this theory to hydrodynamical problems in unbounded (cylindrical) domains.
Such extension was problematic during the long time since even the well-posedness of the Navier-Stokes equation
in 2D unbounded domains in the proper classes of spatially non-decaying
solutions

was not known. The situation is changed now due to recent results on the
solvability and dissipativity of 2D and 3D Navier-Stokes equations in cylindrical domains.

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__ ____More information about PhD studies
at the department and links to application forms are posted at the ____departmental PhD
programme pages____. ____For
informal enquiries, please contact ____Dr Sergey Zelik____ or ____Dr Gianne Derks____ (PhD
admissions tutor)____. __