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Cauchy Problem for Cylinder-like Capillary Jets

Haocheng Yang

Abstract

The motion of liquid jets plays an important role in physics and engineering, and needs rigorous mathematical investigations. Recently, Huang-Karakhanyan proved the first local well-posedness in Sobolev spaces for axisymmetric jets. In this paper, we will extend this result to general jets, namely without any axisymmetry condition.

Cauchy Problem for Cylinder-like Capillary Jets

Abstract

The motion of liquid jets plays an important role in physics and engineering, and needs rigorous mathematical investigations. Recently, Huang-Karakhanyan proved the first local well-posedness in Sobolev spaces for axisymmetric jets. In this paper, we will extend this result to general jets, namely without any axisymmetry condition.

Paper Structure

This paper contains 59 sections, 83 theorems, 584 equations.

Table of Contents

  1. Introduction
  2. History of the problem
  3. Setting of the problem
  4. Cylinder-like jets
  5. Boundary conditions
  6. Impact of gravity
  7. Craig-Sulem-Zakharov formulation
  8. Main results
  9. Idea of the proof
  10. Paralinearization of $G(\eta)$.
  11. Paralinearization of $N$, $H$.
  12. Symmetrization.
  13. Cauchy problem.
  14. Continuity of the solution.
  15. Plan of the paper
  16. ...and 44 more sections

Key Result

Theorem 1.1

Let $(\eta_0-R,\psi_0) \in H^{s+\frac{1}{2}}(\mathbb{T}\times\mathbb{R}) \times H^s(\mathbb{T}\times\mathbb{R})$ with $s>3$. Assume that $\eta_0$ satisfies the hypotheses hyp-intro:bounds and hyp-intro:perturb, or hyp-intro:bounds and hyp-intro:period for periodic case. Then there exists $T>0$, such Moreover, the hypotheses hyp-intro:bounds, hyp-intro:perturb or hyp-intro:period are preserved for

Theorems & Definitions (150)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Lemma 2.6
  • ...and 140 more