The main research aim within the COE-program is the further development of the mathematical theory for the Navier-Stokes equations, the case of incompressible Newtonian fluids. Among the possible directions are: local regularity analysis for weak solutions to the Navier-Stokes equations, existence of strong solutions to initial boundary value problems for the Navier-Stokes equations, uniqueness of weak solutions to initial boundary value problems for the Navier-Stokes equations. In particular, it is planned to develop the technique based on the reduction of local regularity analysis to various backward uniqueness problems for the heat operator with variable lower order terms. It is also supposed to establish new sufficient conditions in terms of the pressure that provide smoothness of weak solutions. Similar problems will be considered in the theory of viscous non-Newtonian fluids.
Another research aim is the mathematical theory of phase equilibrium in elastic solids: quasi-convex envelopes of corresponding non-convex energy functionals and regularity for their minimizers. It is also planned to investigate topological properties of free boundaries separating microstructure zones from pure phase zones.