I will discuss some theoretical and numerical results on the stability properties of a dynamical system modeled by a nonlinear wave equation with a "degenerate" damping. The coefficient of the dissipation term is proportional to the amplitudes, hence its support depends on the geometry of the solution. This feature substantially complicates the stability analysis even…

I will explain some interesting selection problems in nonlinear PDEs. The basic question is about how to select one good solution out of many reasonable ones. A question of this type led to the whole theory of viscosity solutions in 1980s. Then I will focus on the vanishing discount problem and describe the main results,…

We visit several models of polymer flooding in reservoir simulation. A special common feature shared by the models, i.e., the thermo-dynamics is decoupled from the hydro-dynamics, leads to a scalar conservation law with discontinuous flux. We discuss solution of Riemann problems as the vanishing viscosity limit. In particular, we show by counter examples that there exists…