Quasiperiodic Solutions of the Generalized SQG Equation

New, broadly applicable, parameter-free techniques for constructing stable quasiperiodic solutions of quasilinear evolution equations

This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations. The authors implement an ingenious scheme for building global quasiperiodic solutions without relying on external parameters, instead exploiting 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 pseudodifferential calculus, and provides tools that extend beyond the specific SQG context and may prove useful for other evolution equations. Specifically, the authors establish the existence of quasiperiodic patch solutions for the generalized Surface Quasi-Geostrophic (SQG) equation across the parameter range $\alpha \in (1,2)$, in a neighborhood of the disk solution. These solutions exist globally in time without developing singularities, which sheds light on 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. And 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.