Castro, Córdoba, Fefferman, Gancedo and Gómez-Serrano (J. Math. Phys. 53, 115622, 2012) proved that surface tension does not prevent finite-time splash or splat self-intersection of the free surface for the two-dimensional one-phase Euler equations, by solving the equations in a domain where the self-intersecting interface has been desingularized by the conformal map P(w) = (tan(w/2))^{1/2}. Their local theory is complete only in the small-surface-tension, Rayleigh–Taylor-positive regime: the sign-free existence theorem holds for every fixed surface-tension coefficient τ > 0 but with an existence time depending on τ, while the τ-uniform theorem requires strict positivity of the Rayleigh–Taylor function, a condition their own splash data lose at larger τ. The open problem is the completion of that quantitative local theory: existence times (or sharp degeneration results) uniform in τ without Rayleigh–Taylor positivity, the sharp τ-dependence of the sign-free energy estimates, and control of the minimal distances from the desingularized curve to the branch points of P. No resolution is known in the literature: every τ-uniform result assumes Rayleigh–Taylor positivity, and none treats the desingularized system's degenerate regimes.

Background

The water-wave problem asks for the motion of an inviscid, incompressible fluid occupying a planar domain Ω(t) whose boundary ∂Ω(t) is a free surface, parametrized by a curve z(α,t), with vacuum (zero-density air) above; this is the one-phase free-boundary Euler problem. Without surface tension the pressure vanishes on the free surface; with surface tension it obeys the Laplace–Young condition, which states that the pressure on the interface is proportional to its signed curvature κ, namely −p = τκ, where τ > 0 is the surface-tension coefficient (equation (I.5) of the source paper writes −p = (τ/2) z_αα · z_α^⊥/|z_α|³ ≡ (τ/2)K; the τ used throughout this task is one half of the source's τ in (I.5), the factor 1/2 being the entire normalization difference).

Two structural conditions dominate the local theory. First, the arc-chord condition |z(α,t) − z(β,t)| ≥ c_AC|α − β| for some constant c_AC > 0 rules out self-intersection and self-touching of the interface; it is the hypothesis under which classical local well-posedness was proved (Wu 1997 and subsequent work). Second, without surface tension one needs the Taylor sign condition on the Rayleigh–Taylor function σ = −∂_n p, the negative normal derivative of the pressure evaluated at the interface: strict positivity of σ is the stability condition preventing the Rayleigh–Taylor instability, in which a heavier fluid accelerates into a lighter one. With surface tension the local problem is well posed without any sign condition on σ (surface tension regularizes high-frequency growth), but — as the source paper shows numerically — the Rayleigh–Taylor function of splash-type data can change sign at larger τ, so the sign-free regime is genuinely reached, not an artifact.

Castro, Córdoba, Fefferman, Gancedo and Gómez-Serrano constructed the first finite-time self-intersection singularities for the one-phase problem without surface tension (PNAS 2012; Annals of Mathematics 2013): smooth initial data satisfying the arc-chord condition whose interface remains smooth in parameter space up to a time at which it touches itself at a single point (a splash singularity) or along an arc (a splat singularity). The construction runs time backward: one prescribes a self-intersecting splash curve as initial data, solves locally in a domain where the curve has been desingularized, and uses the time reversibility of the Euler equations to obtain smooth data that evolve into the splash.

The source paper (J. Math. Phys. 53, 115622, 2012) carries out the same program with surface tension: for every fixed τ > 0 there are smooth arc-chord data whose interface self-intersects in finite time, so surface tension does not prevent splash or splat. The desingularization is the conformal map

P(w)=(tan⁡w2)1/2,P(w) = \left(\tan\frac{w}{2}\right)^{1/2},

whose square-root branch separates the self-intersection points. The mapped curve z̃ = P(z) satisfies the arc-chord condition, and the evolution closes as a system for (z̃, ω̃), where ω̃ is the vortex-sheet amplitude — the density carried on the interface whose Birkhoff–Rott integral (a singular integral along the interface giving the velocity generated by that vorticity) produces the fluid velocity. The transformed equations carry the weight Q² = |dP/dw(P⁻¹(z̃))|² and modified curvature terms coming from the Laplace–Young condition, and the local theory is built from a priori energy estimates, i.e. bounds on a controlled energy functional whose time derivative is dominated by a power of the energy itself, from which existence and continuation in Sobolev spaces follow.

Two local-existence theorems are proved in this desingularized (tilde) domain. Theorem IV.1 uses an energy requiring no sign of the Rayleigh–Taylor function and works for every τ > 0, but its existence time depends on τ, the estimates containing negative powers of τ. Theorem V.1 gives an existence time independent of τ but requires the Rayleigh–Taylor function σ to be strictly positive initially — a condition the paper verifies for its splash data only when τ is small enough (its Figure 2 shows σ changing sign for the source-normalization values 0.5 and 1, corresponding to τ = 0.25 and τ = 0.5 here). Both energies include the reciprocal minimal distances Σ_{ℓ=0}^{4} 1/m(q^ℓ)(t), where m(q^ℓ)(t) = min_α |z̃(α,t) − q^ℓ| and q⁰ = (0,0), q¹,…,q⁴ = (±1/√2, ±1/√2) are the five singular points of P at which the map degenerates; the estimates hold only while these distances stay bounded away from zero, which on the time scales of the construction is guaranteed only by continuity in time.

The τ-uniform theory is the analytic backbone of the rigorous zero-surface-tension-limit program: Ambrose and Masmoudi (Comm. Pure Appl. Math. 2005) proved for the two-dimensional problem that solutions with surface tension converge, as τ → 0, to solutions without surface tension, under the Taylor sign condition; the source paper's Section V mirrors that structure in the desingularized setting. The present question concerns exactly the regimes where this uniformity and the desingularization itself are not controlled: the sign-changing Rayleigh–Taylor regime, the quantitative τ-dependence of the sign-free theory, and the behavior near the branch points of P.

Problem Statement

Consider the two-dimensional one-phase free-boundary incompressible Euler equations with surface tension (coefficient τ > 0, Laplace–Young condition −p = τκ on the free boundary), in the conformally desingularized formulation of Castro–Córdoba–Fefferman–Gancedo–Gómez-Serrano (J. Math. Phys. 53, 115622, 2012) used to construct splash and splat singularities: the interface is mapped by P(w)=(tan⁡(w/2))1/2P(w)=(\tan(w/2))^{1/2} to a curve z~=P(z)\tilde z=P(z) satisfying the arc-chord condition, and the evolution of (z~,ω~)(\tilde z,\tilde\omega) is closed by a priori energy estimates in the desingularized domain. Establish the parts of the quantitative local theory of this system left uncovered by that paper, in the regimes where the Rayleigh–Taylor function and the desingularization degenerate:

  1. (Rayleigh–Taylor sign-changing regime) Prove a priori estimates and a local existence interval for the desingularized system that is uniform in the surface-tension coefficient τ\tau without assuming strict positivity of the Rayleigh–Taylor function σ(α,t)\sigma(\alpha,t) — in particular for splash-type initial data such as the paper's examples, for which σ\sigma changes sign at larger τ\tau — or, if such uniformity fails, prove sharp results quantifying how any τ\tau-free existence time must degenerate (for example growth or ill-posedness bounds as τ\tau grows past the sign change, in the regime where σ\sigma is somewhere negative).

  2. (Sharp τ\tau-dependence) Quantify the dependence on τ\tau of the existence time furnished by the Rayleigh–Taylor-sign-free energy (the Section-IV-type energy, whose estimates involve negative powers of τ\tau), and determine sharp uniform-in-τ\tau estimates — and, where possible, the limits τ→0\tau\to0 or τ→∞\tau\to\infty — for classes of initial data beyond the small-τ\tau, Rayleigh–Taylor-positive regime covered by the paper's Section V.

  3. (Desingularization degeneracies) (3a) Control the minimal distances m(qℓ)(t)=min⁡α∣z~(α,t)−qℓ∣m(q^\ell)(t)=\min_\alpha|\tilde z(\alpha,t)-q^\ell| from the desingularized curve to the five singular points qℓq^\ell of PP beyond the short-time regime guaranteed by continuity of the solution, either by a quantitative lower bound valid on time scales relevant to splash/splat formation or by a construction of data for which these distances degenerate there; and (3b) characterize the behavior of solutions of the desingularized system when the mapped curve approaches these branch points, in particular what replaces the energy estimates there.

The common objective is a complete, τ\tau-quantified well-posedness theory of the desingularized capillary system that governs splash and splat formation, in the parameter and geometric regimes not covered by the 2012 analysis. The sign-changing regime of item 1 is the special case of the τ\tau-uniform question of item 2 in which the Rayleigh–Taylor function σ\sigma changes sign.

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