Stationary Vortex Patches for the QGSW Equations via Bifurcation

Speaker: Vittorio Baroncini ，University of Seville

Inviter: 薛志龙

Title: Stationary Vortex Patches for the QGSW Equations via Bifurcation

Language: English

Time & Venue: 2026.09.10   16:00-17:00    Zoom: 822 2162 8174    Passcode: 629216

Abstract: We construct non-trivial, m-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 λ>0, we prove that m-fold symmetric stationary patches bifurcate from an annular state for all sufficiently large m. The corresponding inner radii satisfy 1−bm,λ∼β(λ)m as m→∞. Alternatively, keeping the inner radius b fixed, we obtain the complementary asymptotic behavior for the sequence of bifurcating inverse Rossby radii λm,b: λm,b∼2(1−b2)mlogm as m→∞.