We consider the spreading of a droplet on a planar surface covered
with random obstacles. Assuming the obstacles are stationary ergodic
and taller than the droplet, we show that the homogenized limit is
described by a droplet spreading on a flat surface but with a reduced
front speed and surface tension. This is joint work with Inwon Kim.

In the talk we address a system of PDEs describing an
interaction between an incompressible fluid and an elastic
body. The fluid motion is modeled by the Navier-Stokes
equations while an elastic body evolves according to an
linear elasticity equation. On the common boundary, the
velocities and stresses are matched. We discuss available
results on local well-posedness and prove new existence and
uniqueness results with the velocity and the displacement
belonging to low regularity spaces.

The results are joint with A. Tuffaha.

In most practical applications of fluid mechanics, it is the interaction of the fluid with the boundary that is most critical to understanding the behavior of the fluid. Physically important parameters, such as the lift and drag of a wing, are determined by the sharp transition the air makes from being at rest on the wing to flowing freely around the airplane near the wing. Mathematically, the behavior of such flows are modeled by the Navier-Stokes equations. In this talk, I will discuss the asymptotic behavior of solutions to the Navier-Stokes equations at small viscosity under various boundary conditions.

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. In this lecture I will discuss the discrete data assimilation for the incompressible two-dimensional Navier-Stokes
equations. In both cases we obtain bounds on the time interval between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.