[Paper Review] Splash singularities for the one-phase Muskat problem in stable regimes

Ángel Castro, Diego Córdoba+2 more|arXiv (Cornell University)|Nov 29, 2013

Navier-Stokes equation solutions18 references4 citations

TL;DR

This paper establishes the first finite-time splash singularity for the one-phase Muskat problem in a stable regime, where a smooth fluid interface collapses at a single point due to two distinct fluid particles colliding. Using a transformed velocity field and conformal mapping, the authors prove that under the Rayleigh-Taylor condition, such singularities can form despite the parabolic nature of the system, which prevents backward time evolution.

ABSTRACT

This paper shows finite time singularity formation for the Muskat problem in a stable regime. The framework we found is with a dry region, where the density and the viscosity are set equal to $0$ (the gradient of the pressure is equal to $(0,0)$) in the complement of the fluid domain. The singularity is a splash-type: a smooth fluid boundary collapses due to two different particles evolve to collide at a single point. This is the first example of a splash singularity for a parabolic problem.

Motivation & Objective

• To demonstrate the existence of finite-time splash singularities in the one-phase Muskat problem under stable, parabolic dynamics.
• To overcome the challenge of backward time inapplicability in parabolic systems, unlike previous work on water waves.
• To construct initial data satisfying the Rayleigh-Taylor condition that lead to interface self-intersection at a single point.
• To establish a framework for singularity formation in stable regimes using a transformed velocity field and conformal mapping.

Proposed method

• Transform the Muskat problem using a moving frame to decouple gravity effects, introducing a new velocity field $\underline{v}$, density $\underline{\rho}$, and pressure $\underline{p}$.
• Apply a conformal map to the fluid domain to analyze the free boundary in a transformed 'tilde domain', enabling control over geometric singularities.
• Use Hopf's lemma to show that the normal velocity at the splash point separates the two colliding fluid particles, ensuring inward collapse.
• Construct initial data such that the interface is smooth but evolves to self-intersect at a single point $z(\alpha_1) = z(\alpha_2)$ at finite time $T_*$.
• Verify the Rayleigh-Taylor condition holds throughout the evolution, ensuring the regime remains stable and parabolic.
• Employ the identity $\partial_t \underline{\rho} + \underline{v} \cdot \nabla \underline{\rho} = 0$ and $\underline{\mu} \underline{v} = -\nabla \underline{p}$ to maintain the structure of the Muskat equations in the new frame.

Experimental results

Research questions

• RQ1 Can splash singularities form in the one-phase Muskat problem under stable, parabolic dynamics?
• RQ2 Is it possible to construct such singularities despite the inability to solve the parabolic system backward in time?
• RQ3 How can the Rayleigh-Taylor condition be preserved while achieving interface self-intersection at a single point?
• RQ4 What transformation allows the analysis of splash singularities in a stable Muskat regime?
• RQ5 Can conformal mapping techniques used in water wave problems be adapted to the Muskat setting to prove singularity formation?

Key findings

• The paper constructs the first example of a splash singularity for a parabolic PDE, specifically the one-phase Muskat problem in a stable regime.
• The singularity occurs at a finite time $T_*$ where two distinct points on the smooth interface collide at a single spatial point.
• The Rayleigh-Taylor condition is satisfied throughout the evolution, ensuring the stability of the regime and the parabolic nature of the system.
• The velocity field in the transformed frame ensures that the normal components at the two colliding points point inward, enabling the collapse.
• The use of a moving frame transformation allows the system to be recast in a form where backward evolution is not required, circumventing the parabolic time irreversibility.
• The analysis confirms that the singularity is of the 'splash' type (point self-intersection), not 'splat' (arc self-intersection), distinguishing it from previous results in water wave systems.