The Hele--Shaw problem and area-preserving curve-shortening motions (Q687185)

The author studies the following Hele-Shaw problem: Given a simply connected curve in a smooth bounded domain, find the motion of the curve such that its normal velocity equals the normal derivative jump of a function which is harmonic in the complement of the curve in the domain and whose boundary value on the curve equals its curvature. It is shown that a weak solution exists locally in time; if the initial curve is close to a circle, global existence follows. In the latter case, it is proved that the global solution converges to a circle exponentially fast. The main new idea in the existence proof is to regularize the boundary condition involving the normal velocity of the curve by a term involving the second derivative of the curvature with respect to the arc length. For the regularized system existence can be shown using Schauder's fixed point theorem. Then a priori estimates and compactness arguments yield the existence of a sequence of regularized solutions that converge to a solution of the original Hele-Shaw problem.