Talk by Anna Abbatiello

The next seminar “Modelling of materials – theory, model reduction and efficient numerical methods” will take place on Wednesday from 9:00 till 10:00 in the room K3. The talk will be given by Anna Abbatiello. Please see the details below.

Speaker: Anna Abbatiello
Title: On the exponential decay in time of solutions to a generalized Navier-Stokes-Fourier system

Abstract: We consider a flow of non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the spatially inhomogeneous Dirichlet boundary condition for the temperature. The ultimate goal is to show that the fluid converges to equilibrium as time tends to infinity. However, to justify such result, one needs to deal with very special inequalities and very special test functions, which are typically not admissible on the level of weak solutions. We assume no external body forces. For the power-law like models with the power law index bigger than 11/5 in three dimensions, we identify a class of solutions fulfilling the entropy equality and converging to the equilibria exponentially in a proper metric. In fact, we show the existence of a Lyapunov functional for the problem. Consequently, the steady solution is nonlinearly stable and attracts all suitable weak solutions. This is a joint work with Miroslav Bulíček and Petr Kaplický.