No-splash theorems for fluid interfaces.

The article (1) by Gancedo and Strain in PNAS studies how singularities may develop in the initially smooth interfaces separating two or more incompressible fluids. The fluids and interfaces are assumed to evolve by either of the two standard systems of equations from fluid mechanics, namely the surface quasi-geostrophic (SQG) sharp front equation (2) or the Muskat equation (3). Gancedo and Strain prove that initially smooth fluid interfaces evolving by either of those two equations cannot form a splash singularity (4). Let us first describe the SQG sharp front equation. We consider a temperature θðx; tÞ depending on time t and position x= ðx1; x2Þ in the plane. The temperature is carried along by an incompressible fluid whose velocity at position x and time t is uðx; tÞ. That is ð∂t + u ·∇xÞθðx; tÞ= 0: