My current research concerns
global existence for arbitrary nonzero surface tension of
bubbles in a Hele-Shaw cell.
Without imposed pressure gradient or side walls, the circular bubble is shown to be
asymptotically stable to all sufficiently smooth initial perturbation.
For the bubbles translating in the presence of a pressure, when the cell width to bubble size is sufficiently large,
we show that a unique steady translating near-circular bubble symmetric
about the channel centerline exists, where the bubble translation speed in
the laboratory frame is found
as part of the solution.
We prove
global existence for symmetric sufficiently smooth initial conditions
close to this shape and show that the steady translating
bubble solution is an attractor within this class of disturbances.
In the absence of side walls, we prove stability of the steady
translating circular bubble without restriction
on symmetry of initial conditions.
These results hold for any
nonzero surface tension despite the fact that a local planar approximation
near the front of the bubble would suggest
Saffman-Taylor instability. An important element of the proof was
the introduction of a weighted Sobolev norm that
accounts for stabilization due to advection of disturbances
from the front to the back of the bubble.
We exploit a boundary integral approach
that is particularly suitable for analysis of nonzero viscosity ratio
between fluid inside and outside the bubble.