Classical solvability of the radial viscous fingering problem in a Hele–Sha cell with surface tension

We discuss one-phase radial viscous fingering problem in a Hele–Sha cell with surface tension, which is a nonlinear problem with a free boundary for elliptic equations. Unlike the Stefan problem for heat equation Hele–Sha problem is of hydrodynamic type. In this paper the classical solvability of one-phase Hele–Sha problem with radial geometry is established by applying the same method as for the Stefan problem and justifying the vanishing the coefficient of the time-derivative in a parabolic equation.