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A priori estimates for the motion of charged liquid drop: A dynamic approach via free boundary Euler equations

Vesa Julin, Domenico Angelo La Manna

Abstract

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one expects the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness, regularity and the formation of singularities of the solution. Our main theorem roughly states that if the flow remains C^{1,α}-regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., C^{\infty} in time and space, assuming that the initial data is smooth. Due to the appearance of Taylor cones we expect the C^{1,α}-regularity assumption to be optimal, while the Lipschitz-regularity assumption on the velocity is standard in the classical theory of the Euler equations. We also quantify the C^{\infty}-estimate via high order energy estimates. This result is new also for the Euler equations with free boundary without the electric field. We point out that we do not consider the problem of existence in this paper. It will be studied in forthcoming work.

A priori estimates for the motion of charged liquid drop: A dynamic approach via free boundary Euler equations

Abstract

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one expects the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness, regularity and the formation of singularities of the solution. Our main theorem roughly states that if the flow remains C^{1,α}-regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., C^{\infty} in time and space, assuming that the initial data is smooth. Due to the appearance of Taylor cones we expect the C^{1,α}-regularity assumption to be optimal, while the Lipschitz-regularity assumption on the velocity is standard in the classical theory of the Euler equations. We also quantify the C^{\infty}-estimate via high order energy estimates. This result is new also for the Euler equations with free boundary without the electric field. We point out that we do not consider the problem of existence in this paper. It will be studied in forthcoming work.
Paper Structure (19 sections, 39 theorems, 751 equations)

This paper contains 19 sections, 39 theorems, 751 equations.

Table of Contents

  1. Introduction and the main result
  2. State-of-the-art
  3. Statement of the Main Theorem
  4. Overview of the proof and the structure of the paper
  5. Notation and Preliminary results
  6. Half-integer Sobolev spaces
  7. Half-integer Sobolev spaces on a surfaces
  8. Geometric Preliminaries
  9. Functional and geometric inequalities
  10. Elliptic estimates for vector fields and functions
  11. Regularity estimates for vector fields
  12. Regularity estimates for functions
  13. Useful formulas
  14. Estimation of the error terms
  15. First regularity estimates
  16. ...and 4 more sections

Key Result

Proposition 2.1

Let $m \in \mathbb N$, with $m\geq 2$, and let $\Omega$ be a smooth domain which is uniformly $C^{1,\alpha}(\Gamma)$-regular and satisfies eq:notationhp. Then there is an extension operator $T:H^{m}(\Omega)\to H_0^{m} (\mathbb R^3)$ such that

Theorems & Definitions (82)

  • Proposition 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Corollary 2.9
  • ...and 72 more