中国科学院数学与系统科学研究院
数学研究所
数学科学全国重点实验室
偏微分方程讨论班
Speaker: Vittorio Baroncini (University of Seville)
Inviter: 薛志龙
Language: English
Title:Stationary Vortex Patches for the QGSW Equations via Bifurcation
Time & Venue: 2026年9月10日（星期四）16:00-17:00 Zoom: 822 2162 8174 Passcode: 629216
Abstract:We construct non-trivial, 
-fold symmetric, doubly-connected stationary vortex patches for the Quasi-Geostrophic Shallow-Water (QGSW) equations. The existence of these solutions is established through a local bifurcation framework based on the Crandall--Rabinowitz theorem, using either the inner radius of an annulus or the inverse Rossby radius as the bifurcation parameter.
A central element of the analysis relies on modified Bessel functions, which naturally arise in the spectral study of the linearized operator. Their detailed asymptotic expansions, derivatives, recurrence formulas, and monotonicity properties play a key role in identifying the bifurcation points and checking the transversality condition.
For any fixed inverse Rossby radius 
, we prove that 
-fold symmetric stationary patches bifurcate from an annular state for all sufficiently large 
. The corresponding inner radii satisfy
Alternatively, keeping the inner radius 
fixed, we obtain the complementary asymptotic behavior for the sequence of bifurcating inverse Rossby radii 
:
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