Global Solutions for the Generalized SQG Patch Equation

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter α ∈ ( 1 , 2 ) . The cases α = 0 and α = 1 correspond to 2d Euler and SQG respectively, and our choice of the parameter α results in a velocity more singular than in the SQG case. Ou...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2019-09, Vol.233 (3), p.1211-1251
Main Authors: Córdoba, Diego, Gómez-Serrano, Javier, Ionescu, Alexandru D.
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Fluid- and Aerodynamics
Mathematical and Computational Physics
Parameters
Physics
Physics and Astronomy
Theoretical
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Summary:We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter α ∈ ( 1 , 2 ) . The cases α = 0 and α = 1 correspond to 2d Euler and SQG respectively, and our choice of the parameter α results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-019-01377-6