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On the global well-posedness of interface dynamics for gravity Stokes flow

Francisco Gancedo, Rafael Granero-Belinchón, Elena Salguero

Abstract

In this paper we establish the global-in-time well-posedness for an arbitrary $C^{1+γ}$, $0<γ<1$, initial internal wave for the free boundary gravity Stokes system in two dimensions. This classical well-posedness result is complemented with a weak solvability result in the case of $C^γ$ or Lipschitz interfaces. Furthermore, we also propose and study several one-dimensional models that capture different features of the full internal wave problem for the gravity Stokes system and show that all of them present finite time singularities. This fact evidences the fine structure of the non-linearity in the full system that allows for the free boundary problem to be globally well-posed while simplifications of it blow-up in finite time.

On the global well-posedness of interface dynamics for gravity Stokes flow

Abstract

In this paper we establish the global-in-time well-posedness for an arbitrary , , initial internal wave for the free boundary gravity Stokes system in two dimensions. This classical well-posedness result is complemented with a weak solvability result in the case of or Lipschitz interfaces. Furthermore, we also propose and study several one-dimensional models that capture different features of the full internal wave problem for the gravity Stokes system and show that all of them present finite time singularities. This fact evidences the fine structure of the non-linearity in the full system that allows for the free boundary problem to be globally well-posed while simplifications of it blow-up in finite time.
Paper Structure (5 sections, 11 theorems, 140 equations)

This paper contains 5 sections, 11 theorems, 140 equations.

Table of Contents

  1. Introduction and main results
  2. Global well-posedness of the internal wave for the gravity Stokes system
  3. Global well-posedness vs. finite time singularities in several one-dimensional models of the internal waves for the gravity Stokes system
  4. Global well-posedness
  5. Finite-time singularities

Key Result

Theorem 1

Let $z_0(\alpha) \in C^{1+\gamma}(\mathbb T)$ be the initial data satisfying the arc-chord condition for the two-phase Stokes problem Stokes. Then, there exists a solution for any $T>0$.

Theorems & Definitions (21)

  • Theorem 1: Global well-posedness of two-phase Stokes
  • Theorem 2: Global well-posedness vs. finite time singularities for 1D models
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['teo1']}
  • Corollary 1
  • proof
  • Theorem 3: Existence of solutions for small data
  • ...and 11 more