( Faculty of Mathematics and Physics, Charles University in Prague )
Regularity and uniqueness of non-Newtonian binary fluid mixtures
Abstract:
We consider a binary fluid mixture, described by the Ladyzhenskaya type model for the velocity, and coupled with the Cahn-Hilliard equation for the order parameter, confined to a bounded three-dimensional domain.
The system is complemented with some reasonable boundary condition, e.g. zero Dirichlet / Neumann for velocity / order parameter. The point is that we will mainly work with (fractional) time regularity of solutions, so that the results hold -- with suitable modifications -- for any boundary condition.
It is well-known that for $p>=11/5$ (the parameter of the growth of stress tensor w.r. to symmetric velocity gradient) weak solutions to Ladyzhenskaya model exist globally, and satisfy the energy equality. Recently, it has been shown by Bulíček, Ettwein, Kaplický and the author, that for $p>11/5$, any weak solution is regular enough to ensure uniqueness; moreover, there exists a finite-dimensional exponential attractor. In our talk, we will extend the argument -- which uses iterative scheme of improving time regularity in fractional Nikolskii spaces -- to the case of binary mixtures.