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Solvability for the Ginzburg-Landau equation linearized at the degree-one vortex

Manuel del Pino, Rowan Juneman, Monica Musso

Abstract

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{iθ}$. Using explicit representation formulae for the Fourier modes in $θ$, we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions.

Solvability for the Ginzburg-Landau equation linearized at the degree-one vortex

Abstract

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution . Using explicit representation formulae for the Fourier modes in , we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions.

Paper Structure

This paper contains 9 sections, 25 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Fourier mode k=0
  3. Fourier mode k=1
  4. Solutions of the homogeneous problem: Fourier mode k=1
  5. Solutions of the inhomogeneous problem: Fourier mode k=1
  6. Fourier modes k>=2
  7. Solutions of the homogeneous problem: Fourier modes k>=2
  8. Solutions of the inhomogeneous problem: Fourier modes k>=2
  9. Summing the Fourier modes

Key Result

Theorem 1.1

There exists a constant $C>0$ such that for any $h$ satisfying $\lVert h\rVert_{**}<\infty$ and the orthogonality conditions $\left\langle h,\frac{\partial W}{\partial x_{1}} \right\rangle=\left\langle h,\frac{\partial W}{\partial x_{2}} \right\rangle=0$, equation (linearizedeq) has a solution $\phi

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Proposition \ref{['fouriermode0prop']}
  • Proposition 3.1
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • ...and 41 more