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

Dallas Albritton; Zachary Bradshaw

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arXiv · 2011.10856

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

Dallas Albritton·Zachary Bradshaw

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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.

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Dallas Albritton, Zachary Bradshaw. 2020-11-21. Non-decaying solutions to the critical surface quasi-geostrophic equations with symmetries. https://arxiv.org/abs/2011.10856

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