Quasiperiodic solutions of the generalized SQG equation

Abstract

This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations. We implement an elaborate scheme for building global quasiperiodic solutions without relying on external parameters. Instead of relying on artificial external parameters, we exploit the natural structure of initial data to generate families of stable solutions. This approach offers a more robust framework for studying global solutions of quasilinear PDEs. The book combines techniques from KAM theory, a Nash-Moser iteration scheme, and pseudo-differential calculus, and provides tools that extend beyond the specific SQG context and may prove useful for other evolution equations. Specifically, we establish the existence of quasiperiodic patch solutions for the generalized Surface Quasi-Geostrophic equation (SQG) across the parameter range $α\in (1,2)$, in a neighborhood of the disk solution. These solutions exist globally in time without developing singularities, addressing an important question about the behavior of geophysical fluid models. This work provides new insights into global dynamics in a mathematically challenging regime where standard perturbative methods are insufficient. The techniques developed here offer potential applications to other evolution equations in mathematical physics, making this a valuable resource for researchers in partial differential equations, fluid dynamics, and related fields.