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The Dirichlet-Neumann Operator for Taylor's Cone

Yucong Huang, Aram Karakhanyan

Abstract

The aim of this paper is to analyse the Dirichlet-Neumann operator in axially symmetric conical domains. We provide a constructive treatment of the generic singularity at the vertex by using a new coordinate system that maps the conical domain to a strip. Building upon the paradifferential theory, we then establish our main Sobolev estimates. We also find the shape derivative, the linearization formula, and the cancellation property for the Dirichlet-Neumann operator. Our results can be viewed as the first step towards establishing the mathematical framework for the perturbations of Taylor's cone which appears in the jet break-up control.

The Dirichlet-Neumann Operator for Taylor's Cone

Abstract

The aim of this paper is to analyse the Dirichlet-Neumann operator in axially symmetric conical domains. We provide a constructive treatment of the generic singularity at the vertex by using a new coordinate system that maps the conical domain to a strip. Building upon the paradifferential theory, we then establish our main Sobolev estimates. We also find the shape derivative, the linearization formula, and the cancellation property for the Dirichlet-Neumann operator. Our results can be viewed as the first step towards establishing the mathematical framework for the perturbations of Taylor's cone which appears in the jet break-up control.
Paper Structure (37 sections, 30 theorems, 295 equations)

This paper contains 37 sections, 30 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Behaviour in Sobolev spaces
  4. Linearization of DN operator
  5. Cancellation Property
  6. Organization of paper
  7. Dirichlet Boundary Value Problem for Conical Domain
  8. Formulation in the spherical coordinate
  9. Reformulation in the curved strip
  10. Flat Cone Reformulated in the Strip Domain
  11. Solutions as convolution with Poisson kernel
  12. Bounds for conical Legendre function
  13. Sobolev Estimates for Phistar
  14. Elliptic Estimates for Non-Flat Conical Domain
  15. Existence of weak solutions
  16. ...and 22 more sections

Key Result

Theorem 2.1

Denote $\tilde{\eta}\vcentcolon=\eta-\theta_{\ast}$ with $\theta_{\ast}\in(0,\pi)$. Suppose $\tilde{\eta}\in H^{s+\frac{1}{2}}(\mathbb{R})$ with $s>\frac{5}{2}$. Then $\phi\mapsto \mathcal{G}[\eta](\phi)$ exists as a bounded linear map from $H^{m}(\mathbb{R})$ to $H^{m-1}(\mathbb{R})$ for $\frac{1}{ where the functional $\mathfrak{U}_s(\tilde{\eta})$ and $\mathfrak{l}(\tilde{\eta})$ are defined as

Theorems & Definitions (54)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • Proposition 4.4
  • proof
  • ...and 44 more