On the construction of exact solutions to the Muskat problem

To construct exact solutions of the unsteady two-phase Hele-Shaw problem (Muskat problem), a new modification of the S. Howison's method has been proposed. The efficiency of this modification has been demonstrated through the reconstruction of the D. Crowdy's explicit solution for nonstationary elliptic bubble of one viscous fluid surrounded by another viscous fluid. This solution has been found to be in direct relation with the explicit solution for similar one-phase Hele-Shaw problem. Comparison of these solutions has shown that there is a weak regularization when we take into account the second phase: the one-phase solution exists for a finite time while the two-phase solution exists for an infinite time under the same initial conditions.