Hyperbolic evolution equations for moving boundary problems

Short-time existence and uniqueness results in Sobolev spaces are proved fo r Hele-Shaw flow with kinetic undercooling and for Stokes flow without surf ace tension. In both cases, the flow is driven by arbitrarily distributed s ources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sph ere, quasilinearization by equivariance, investigation of the linearization , and Galerkin approximations. Tn the situation discussed here, the lineari zed evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the p roperties of the evolution equations that arise from Hele-Shaw flow and Sto kes how with and without regularization is given.