Nonlinear interpolation and the flow map for quasilinear equations

TL;DR

This paper develops a new interpolation theorem for nonlinear functionals on Banach space scales, enabling automatic continuity results for flow maps in quasilinear PDEs based on existing a priori estimates.

Contribution

It introduces a generalized interpolation theorem applicable to locally defined nonlinear functionals, extending classical results and simplifying the analysis of quasilinear evolution equations.

Findings

Interpolation theorem for nonlinear functionals on Banach scales

Automatic continuity of flow maps from a priori estimates

Extension of classical interpolation results to local, weakly Lipschitz functionals

Abstract

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.