Classical solvability of the radial viscous fingering problem in a Hele–Shaw cell

We discuss a single-phase radial viscous fingering problem in a Hele–Shaw cell, which is a nonlinear problem with a free boundary for an elliptic equation. Unlike the Stefan problem for heat equation Hele–Shaw problem is of hydrodynamic type. In this paper a single-phase Hele–Shaw problem in a radial flow geometry admits a unique classical solution by applying the same method as for Stefan problem and justifying the vanishing the coefficient of the derivative with respect to time in a parabolic equation.