Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé I transcendent

► The paper deals with the famous Hele-Shaw problem of 2D fluid dynamics. ► It presents in premiere a solution for cusp-formation in Hele-Shaw flows. ► This solution preserves the integrability of the dynamics before cusps. ► The solution explains the viscous fingering phenomenon. ► The method introduced in the paper can be used in many other problems.