No-splash theorems for fluid interfaces

Abstract

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 = ðx 1 ; x 2 Þ in the plane. The temperature is carried along by an incompressible fluid whose velocity at position x and time t is uðx; tÞ. That isBecause the fluid is incompressible, the velocity uðx; tÞ arises from a stream function ψ by the standard formula uðx; tÞ = − ∂ψ ∂x 2 ; ∂ψ ∂x 1 :We take the stream function ψ to arise from the temperature θ by the formula ψðx; tÞ = Z R 2 jx − yj −1 θðy; tÞdy:The above system for ðθ; u; ψÞ is called the Suppose we take the temperature θðx; tÞ equal to 1 if x lies inside a closed contour ΓðtÞ and equal to 0 if x lies outside ΓðtÞ. Then the fluid velocity uðx; tÞ diverges logarithmically as x approaches the contour ΓðtÞ. Nevertheless, it is possible to make sense of the SQG equation in this case. This leads to an evolution equation for the contour ΓðtÞ, called the SQG sharp front equation (2).The SQG sharp front equation crudely models the evolution of a cold front on the earth's surface. It also serves as a simplified model of vortex filaments for the 3D Euler equation.Suppose that the initial contour Γð0Þ is smooth. We ask whether the contour ΓðtÞ may develop a singularity at some positive time. If so, then we ask how that singularity might look.Rodrigo (2) proved that an initially smooth contour Γð0Þ, evolving by the SQG sharp front equation, remains smooth up to some positive time τ [that may depend on Γð0Þ] (7). Numerical simulations (8) suggest that an assymptotically self-similar singularity arises in finite time; however, no mathematical rigorous proof of this is known.Next, we describe the Muskat equation in two space dimensions. We consider two or more incompressible, inmiscible fluids in a porous medium (e.g., oil and water in sand). The fluids are separated by one or more interfaces. The same equations that govern the above system also describe the behavior of two or more fluids trapped between two closely spaced parallel vertical plates (a "Hele Shaw cell") (9).Each fluid has its own density and viscosity. We assume here that the fluids all have the same viscosity, but their densities are different.The fluid velocity uðx; tÞ and pressure pðx; tÞ are assumed to satisfy Darcy's law (10), which, in suitable units, asserts that uðx; tÞ = − ∇pðx; tÞ − ð0; ρÞ;where ρ denotes the density of the fluid at position x and time t. We assume also that the pressure pðx; tÞ is continuous across the interface, and the jump in the velocity ...

Large time decay estimates for the Muskat equation

“…For (21), when ν > −d then we require additionally that f 0 ν < ∞, and when ν = −d then we alternatively require f 0 −d,∞ < ∞. The implicit constants in (20) and (21) depend on f 0 s < ∞ and k 0 .…”

“…The implicit constants in (20) and (21) depend on f 0 s < ∞ and k 0 . In (21) the implicit constant further depends on either f 0 ν (when ν > −d) or f 0 −d,∞ (when ν = −d).…”

“…For (21), when ν > −d then we require additionally that f 0 ν < ∞, and when ν = −d then we alternatively require f 0 −d,∞ < ∞. The implicit constants in (20) and ( It can be directly seen from the proof that for (21), when ν > −d then one only needs to assume f 0 s,∞ < ∞ instead of the stronger condition f 0 s < ∞. Also note that it is shown in Proposition 14 and Section 5 that f −d,∞ (t) 1 and we more generally have f s,∞ (t) 1 for ν ≥ −d from (12).…”

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

“…The latter setup in turn admits the well-known water-wave, or air-water, limit, in which the density of the lighter fluid, air, is assumed to be negligible and enters the dynamics only as means of maintaining constant pressure on the water's free surface. There are some similarities between the "splash" and "splat" singularities studied in [12,21,22] (or their backward-time versions) and the focus of this work on the interaction of a fluid's density isoline with rigid boundaries. For instance, the splash example provided in [21], due to the reflectional symmetry across a vertical line, can be viewed as interaction of the free surface with a vertical wall.…”

The domain of parabolicity for the Muskat problem

“…Existence of solutions for nonregular initial data is shown in [5] by means of a fixed point argument. There are various interesting phenomena established for fluids with equal viscosities: global existence of strong and weak solutions for initial data which are bounded by explicit constants [10,27], existence of initial data for which solutions turn over [7][8][9], or the absence of squirt or splash singularities [13,23,25].…”