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The Hele–Shaw problem and area-preserving curve-shortening motions. (English) Zbl 0780.35117
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.
Reviewer: J.Sprekels (Essen)

MSC:
35R35 Free boundary problems for PDEs
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
35R30 Inverse problems for PDEs
35B45 A priori estimates in context of PDEs
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