TY - JOUR
T1 - Global Solutions for the Generalized SQG Patch Equation
AU - Córdoba, Diego
AU - Gómez-Serrano, Javier
AU - Ionescu, Alexandru D.
N1 - Funding Information:
The first two authors were supported in part by the grant MTM2014-59488-P (Spain) and ICMAT Severo Ochoa projects SEV-2011-008 and SEV-2015-556. The first author was supported in part by a Minerva Distinguished Visitorship at Princeton University. The second author was supported in part by an AMS Simons Travel Grant. Part of this work was done while some of the authors were visiting ICMAT and Princeton University, to which they are grateful for their support. The last author was supported in part by NSF Grant DMS-1600028 and by NSF-FRG Grant DMS-1463753.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/9/1
Y1 - 2019/9/1
N2 - 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.
AB - 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.
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U2 - 10.1007/s00205-019-01377-6
DO - 10.1007/s00205-019-01377-6
M3 - Article
AN - SCOPUS:85064344631
SN - 0003-9527
VL - 233
SP - 1211
EP - 1251
JO - Archive for Rational Mechanics and Analysis
JF - Archive for Rational Mechanics and Analysis
IS - 3
ER -