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