Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback. In this talk I discuss some general properties of monotone dynamical systems, especially classical and recent results regarding their generic convergence towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as global attractivity to an equilibrium, switches and oscillations under time delays.

In recent work, we have investigated various aspects of the asymptotic behavior of solutions to systems that are known to describe the behavior of incompressible flows of binary fluids,that is, fluids composed by either two phases of the same chemical species or phases of different composition. We intend to give an overview on the following issues: existence and main properties of (trajectory or global) attractors, exponential attractors, convergence to single equilibria, etc.

In recent work, we have investigated various aspects of the asymptotic behavior of solutions to systems that are known to describe the behavior of incompressible flows of binary fluids, that is, fluids composed by either two phases of the same chemical species or phases of different composition. We intend to give an overview on the following issues: existence and main properties
of (trajectory or global) attractors, exponential attractors, convergence to single equilibria, etc.

In this talk we present a joint work with Lei Zhen of Fudan University.

Let v=v(x, t) be a solution to the 3 d axis symmetric NS.
Let (x_0, t_0) be a point such that the flow speed |v(x_0,t_0)| is comparable to the maximum speed for time t