Inversion of a non-uniform difference operator and a strategy for Nash–Moser

We consider the problem of inverting the linear difference operator $\Delta_\Phi [v] = v \circ \Phi - v$ and obtaining bounds for the inverse operator, where $\Phi$ is a non-uniform shift on the circle. This represents the scalar version of a linearized difference operator arising in the construction of periodic solutions to the compressible Euler equations by Nash–Moser methods. We characterize the degeneracies in the linearized operators, thereby describing the complications that can arise in the application of Nash–Moser iteration to quasilinear problems. There are two cases, resonant and nonresonant, which correspond to the rationality or irrationality of the rotation number of $\Phi$, respectively. We introduce a solvability condition which characterizes the range of the difference operator, and obtain uniform bounds for the inverse operator $\Delta^{-1}_\Phi$ on this range in both cases, but our bounds are not immediately expressible in terms of standard $C^r$ or Sobolev norms. In the resonant case, the bound is in terms of the inverse width of “Arnold tongues”. In the non-resonant case the solvability condition simplifies and we translate our estimate into uniform estimates on Sobolev norms with a uniform loss of derivatives, as required for the Nash–Moser method. Our analysis is based on the introduction of the “ergodic norm”, which in addition provides an effective rate of convergence in the classical ergodic theorem.

Keywords

Nash–Moser, compressible Euler equations, ergodic theorem, Arnold tongues

2010 Mathematics Subject Classification

35L65, 37K55

Full Text (PDF format)

Received 6 January 2022

Accepted 7 June 2022

Published 21 March 2023