Potential theory yields local well-posedness, parabolic smoothing in near-optimal spaces, and exponential stability of equilibria for the capillarity-driven 2D Hele-Shaw problem via a generalized linearized stability principle for abstract quasilinear parabolic equations.
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A potential theory approach to the capillarity-driven Hele-Shaw problem
Potential theory yields local well-posedness, parabolic smoothing in near-optimal spaces, and exponential stability of equilibria for the capillarity-driven 2D Hele-Shaw problem via a generalized linearized stability principle for abstract quasilinear parabolic equations.