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

TL;DR

This paper provides sharp estimates for the inverse of the linearized Ginzburg-Landau operator around a degree-one vortex, facilitating analysis of solutions without orthogonality constraints.

Contribution

It introduces explicit Fourier mode representations and sharp inverse estimates for the linearized operator, advancing understanding of vortex solutions in the Ginzburg-Landau equation.

Findings

Sharp inverse estimates for the linearized operator

Explicit Fourier mode representation formulae

Applicability to non-orthogonal right-hand sides

Abstract

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W(x)=w(r)e^{i\theta}$. Using explicit representation formulae for the Fourier modes in $\theta$, 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.