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.

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).