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Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension

Siddhant Agrawal, Neel Patel

TL;DR

This work addresses the dynamics of a single-phase Hele-Shaw interface with surface tension in the plane, focusing on interfaces that begin with a corner. The authors develop a conformal-coordinate framework to derive a third-order nonlocal self-similar equation dominated by the Hilbert transform, and they first analyze a linear approximation to understand the asymptotics. Using a weighted functional-analytic setup, they prove existence, decay, and regularity for the linear problem via a δ-regularized variational scheme, then extend to the full nonlinear equation through a contraction mapping for small corner perturbations, yielding a family of self-similar solutions that are smooth for $t>0$. The results provide explicit structural insight into corner dynamics under surface tension and establish a rigorous foundation for self-similar free-boundary behavior in Hele-Shaw-type problems.

Abstract

We consider the Hele-Shaw problem with surface tension in an infinite domain. We prove the existence of a family of self-similar solutions. At $t=0$, these solutions have a corner of angle $θ$ with $ 0 < |θ- π| \ll 1$, and for $t>0$, the solutions are smooth.

Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension

TL;DR

This work addresses the dynamics of a single-phase Hele-Shaw interface with surface tension in the plane, focusing on interfaces that begin with a corner. The authors develop a conformal-coordinate framework to derive a third-order nonlocal self-similar equation dominated by the Hilbert transform, and they first analyze a linear approximation to understand the asymptotics. Using a weighted functional-analytic setup, they prove existence, decay, and regularity for the linear problem via a δ-regularized variational scheme, then extend to the full nonlinear equation through a contraction mapping for small corner perturbations, yielding a family of self-similar solutions that are smooth for . The results provide explicit structural insight into corner dynamics under surface tension and establish a rigorous foundation for self-similar free-boundary behavior in Hele-Shaw-type problems.

Abstract

We consider the Hele-Shaw problem with surface tension in an infinite domain. We prove the existence of a family of self-similar solutions. At , these solutions have a corner of angle with , and for , the solutions are smooth.

Paper Structure

This paper contains 13 sections, 27 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Main results
  4. Equation for Self-Similarity
  5. Linear approximation
  6. Derivation of the main equation
  7. Linear Analysis
  8. Existence of a Weak Solution to the Linear Equation
  9. Decay and Gain of Regularity of the Linear Solution
  10. Uniqueness of the Linear Solution
  11. Further Properties of the Linear Solution
  12. The Full Nonlinear Equation
  13. Appendix

Key Result

Lemma 2.1

Let $1<p<\infty$ and let $F(z)$ be a holomorphic function in the lower half plane with $F(z) \to 0$ as $z\to \infty$. Then the following are equivalent:

Theorems & Definitions (49)

  • Lemma 2.1: Ti86
  • Theorem 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • Theorem 5.2: Lax-Milgram
  • Lemma 5.3
  • proof
  • ...and 39 more