Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime. (English) Zbl 1465.76043

Summary: 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.


76E20 Stability and instability of geophysical and astrophysical flows
76E30 Nonlinear effects in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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