Non-decaying solutions to the critical surface quasi-geostrophic equations with symmetries

Abstract

We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying as $|x| \to +\infty$, which leads to ambiguity in the velocity $\vec{R}^\perp \theta$. This ambiguity is corrected by imposing $m$-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential non-uniqueness, due to symmetry-breaking bifurcations, in analogy with work \cite{jiasverakillposed,guillodsverak} on the Navier-Stokes equations.