The Perona-Malik equation is a celebrated example of nonlinear forward-backward diffusion, introduced in the context of image denoising. It can be viewed as the formal gradient-flow of a functional with a convex-concave integrand. In spite of its mathematical ill-posedness, numerical experiments exhibit better than expected behavior of its solutions.
After a general introduction to the equation itself, we present a few approximation schemes, some classical and some more recent. The approximating solutions show distinct behavior on three time scales (called fast, standard, and slow time). We provide a rigorous explanation for the slow time behavior of the different approximations.
In order to carry out this analysis, we prove an abstract result about passing to the limit in gradient-flows (in the more general context of the theory of maximal slope curves in metric spaces). We are guided by the general principle that "the limit of a family of gradient-flows is the gradient-flow of the limiting functional".

Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley-
Leverett equation, a scalar conservation law. In the first part of the talk, we study traveling wave solutions of the equation modfied by the Gray-Hassanizadeh model for rate-dependent capillary pressure. The modfication adds a BBM-type dispersion to the classic equation, giving rise to under-compressive waves. In the second part of the talk, we analyze stability of sharp planar interfaces (corresponding to Lax shocks) to two-dimensional perturbations, which involves a system of partial differential equations. The Safman-Taylor analysis predicts instability of planar fronts, but their
calculation lacks the dependence on saturations in the Buckley-Leverett equation. Interestingly, the dispersion relation we derive leads to the conclusion that some interfaces are long-wave stable and some are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. This is joint work with Kim Spayd and Zhengzheng Hu.

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.