Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the onset of filamentation on two-dimensional vorticity interfaces

Adrian Constantin, David Dritschel, Pierre Germain

Abstract

We study an asymptotic nonlinear model for filamention on two-dimensional vorticity interfaces. Different re-formulations of the model equation reveal its underlying structural properties. They enable us to construct global weak solutions and to prove the existence of traveling waves.

On the onset of filamentation on two-dimensional vorticity interfaces

Abstract

We study an asymptotic nonlinear model for filamention on two-dimensional vorticity interfaces. Different re-formulations of the model equation reveal its underlying structural properties. They enable us to construct global weak solutions and to prove the existence of traveling waves.

Paper Structure

This paper contains 24 sections, 10 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. The equation
  3. Known results
  4. Plan of the article and results obtained
  5. Notations
  6. Structure of the equation
  7. The energy
  8. Form of the filamentation equation in Fourier space
  9. Symmetries and invariants
  10. Local strong solutions
  11. Global weak solutions
  12. Variational aspects
  13. Sign of the energy
  14. Weak compactness and lower semi-continuity of the energy
  15. Constrained minimization
  16. ...and 9 more sections

Key Result

Theorem 3.1

The filamentation equation is locally well-posed in $H^s$ if $s> \frac{3}{2}$. More precisely, if $u_0 \in H^s$, there exists $T>0$ and a unique solution $u \in \mathcal{C}([0,T],H^s)$ to the Cauchy problem

Theorems & Definitions (21)

  • Theorem 3.1
  • proof
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Proposition 5.3
  • ...and 11 more