## Grant IAA100190905
1.1.2009 - 31.12.2011

Grantor: Grant Agency of Czech Academy of Sciences

### Dynamical properties of the Navier-Stokes and related equations

The project is focused on the study of 1) asymptotic and dynamic properties of solutions of the Navier-Stokes (= N-S) equations, 2) flows of a N-S fluid in a channel and 3) stability of a solution to the N-S equations modelling flow around a compact body. Item 1) concerns the expansion of a solution to modes (frequencies) and asymptotic behaviour of the modes, with the accent to the question which modes overrule the others and in which ratio at certain times (especially time tending to infinity). Item 2) is interesting due to the non-standard boundary condition on the outflow of the channel. We will deal with existence, respectively uniqueness of solutions (mainly strong) to the N-S equations and we will also extend the mathematical model and qualitative results to flows of a heat conductive fluid. In item 3), we will focus on sufficient conditions for stability without restriction on the size of a basic flow, using especially spectral properties of an associated linear operator.