Classical Solution to an Interfacial Flow with Kinetic Undercooling in a Time-Dependent Gap Hele-Shaw Cell

Hele-Shaw cells where the top plate is moving uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This paper focuses on an interfacial flow with kinetic undercooling regularization in a radial Hele-Shaw cell with a time dependent gap \(b(t)\) . We obtain the local existence of classical solution of the moving boundary problem when the initial data is close to a circle. The methodology is to use complex analysis and reduce the free boundary problem to a Riemann-Hilbert problem and an abstract Cauchy-Kovalevskaya evolution problem, then apply Nash-Moser iteration.