Electronic ISBN:  9781470414818 
Product Code:  MEMO/228/1069.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 228; 2014; 108 ppMSC: Primary 58; Secondary 35;
The authors consider the Schrödinger Map equation in \(2+1\) dimensions, with values into \(\mathbb{S}^2\). This admits a lowest energy steady state \(Q\), namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that \(Q\) is unstable in the energy space \(\dot H^1\). However, in the process of proving this they also show that within the equivariant class \(Q\) is stable in a stronger topology \(X \subset \dot H^1\).

Table of Contents

Chapters

1. Introduction

2. An outline of the paper

3. The Coulomb gauge representation of the equation

4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces

5. The linear $\tilde H$ Schrödinger equation

6. The time dependent linear evolution

7. Analysis of the gauge elements in $X,LX$

8. The nonlinear equation for $\psi $

9. The bootstrap estimate for the $\lambda $ parameter.

10. The bootstrap argument

11. The $\dot H^1$ instability result


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The authors consider the Schrödinger Map equation in \(2+1\) dimensions, with values into \(\mathbb{S}^2\). This admits a lowest energy steady state \(Q\), namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. The authors prove that \(Q\) is unstable in the energy space \(\dot H^1\). However, in the process of proving this they also show that within the equivariant class \(Q\) is stable in a stronger topology \(X \subset \dot H^1\).

Chapters

1. Introduction

2. An outline of the paper

3. The Coulomb gauge representation of the equation

4. Spectral analysis for the operators $H$, $\tilde H$; the $X,L X$ spaces

5. The linear $\tilde H$ Schrödinger equation

6. The time dependent linear evolution

7. Analysis of the gauge elements in $X,LX$

8. The nonlinear equation for $\psi $

9. The bootstrap estimate for the $\lambda $ parameter.

10. The bootstrap argument

11. The $\dot H^1$ instability result