Elliptic and Parabolic Equations - Abstract

Ehrnström, Mats

Well-posedness, instabilities and bifurcation for the flow in a rotating Hele--Shaw cell

We study the motion of an incompressible fluid located in a Hele--Shaw cell rotating at constant speed in the horizontal plane. This is a time-dependent moving-boundary problem, which we recast as an abstract Cauchy problem. Using elliptic theory in combination with theory for analytic semi-groups on suitable spaces, local existence and uniqueness of solutions is proved.
Another topic of interest is time-independent solutions. In this context, we show that there is exactly one rotationally invariant equilibrium, and that it is unstable. There are, however, other time-independent solutions: with the aid of bifurcation theory we establish the existence of global branches of stationary fingering patterns. Further properties of the solutions along those branches can then be deduced.
The talk is based on joint work with J. Escher and B.-V. Matioc.