| Abstract: |
| Of concern is the moving boundary problem of a two-phase potential flow
of two fluids with possible different densities and viscosities. Such problems
are known as Muskat problems or two-phase Hele-Shaw flows. Due to the
moving interfaces these problems are intrinsically nonlocal and highly nonlinear.
A criterion is presented, known as the generalised Rayleigh-Taylor
condition, which guarantees that for large classes of initial data these problems
are classically well-posed, possibly on a finite time interval only. Away from the Rayleigh-Taylor
regime the system becomes unstable and finger-shaped unstable steady states can occur.
A thin film approximation is also discussed. Here the dynamical behaviour is different: global
weak solutions exist for any square integrable non-negative initial configuration. In addition,
the flat steady state is globally stable in the class of weak solutions. |
|