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arXiv · 1910.06955

Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime

Aynur Bulut·Hongjie Dong
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Abstract

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional Laplacian, the nonlinear instability is expressed in terms of the scaling invariant norm, while we establish stronger instability claims in the setting of logarithmically supercritical dissipation. A key tool in treating the logarithmically supercritical setting is a global well-posedness result for the forced equation, which we prove by adapting and extending recent work related to nonlinear maximum principles. We believe that our proof of global well-posedness is of independent interest, to our knowledge giving the first large-data supercritical result with sharp regularity assumptions on the forcing term.

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Aynur Bulut, Hongjie Dong. 2019-10-15. Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime. https://arxiv.org/abs/1910.06955

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