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

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 $\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }$, 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.