Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem

Abstract

Significance The formation of singularities for the evolution of the interphase between fluids with different characteristics is a fundamental problem in mathematical fluid mechanics. These contour dynamics problems are given by fundamental fluid laws such as Euler’s equation, Darcy’s law, and surface quasi-geostrophic (SQG) equations. This work proves that contours cannot intersect at a single point while the free boundary remains smooth—a “splash singularity”—for either the sharp front SQG equation or the Muskat problem. Splash singularities have been shown for water waves. The SQG equation has seen numerical evidence of single pointwise collapse with curvature blow-up. We prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse, confirming the numerical experiments.