The regularity of the boundary of vortex patches for the quasi-geostrophic shallow-water equations

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW equations converge locally in time to the corresponding Euler solutions as $\varepsilon \to 0$ in little Hölder spaces.

💡 Research Summary

The paper studies the quasi‑geostrophic shallow‑water (QGSW) equations, a two‑dimensional active‑scalar model that generalizes the 2‑D Euler equations by incorporating the Rossby radius parameter ε⁻¹. The authors focus on two central problems: (i) the persistence of boundary regularity for vortex‑patch solutions, and (ii) the rigorous convergence of QGSW solutions to Euler solutions as ε→0.

First, the authors introduce the velocity kernel K defined by the modified Bessel functions K₀ and K₁:
 v = K * q, K(x)= (ε/2π) x⊥/|x| K₁(ε|x|).
They collect precise asymptotics for Kₙ(z) both near the origin and at infinity, and establish key estimates such as |K(x)| ≤ C|x|⁻¹, |∇K(x)| ≤ C|x|⁻², and a zero‑mean property on spheres. These estimates allow them to treat the convolution operators T f = K * f and S f = p.v. ∂ᵢK * f as bounded maps on Hölder spaces C^γ and on the little Hölder space c^γ_c.

Using the particle‑trajectory method, they write the flow map X(α,t) solving
 ∂ₜX(α,t)=∫ K(X(α,t)−X(α′,t)) q₀(α′) dα′, X(α,0)=α.
The right‑hand side is a locally Lipschitz map on the Banach space C^{1,γ}, thanks to the kernel bounds and the zero‑mean property. Applying Picard–Lindelöf, they obtain a unique local flow map; the kernel’s decay and the conservation of q along trajectories then extend the solution globally. Consequently, for any initial datum q₀∈C^γ_c, the solution q(x,t)=q₀(X⁻¹(x,t)) stays in C^γ_c for all times.

When q₀ is the characteristic function χ_{Ω₀} of a bounded domain with C^{1,γ} boundary, the flow map’s C^{1,γ} regularity implies that Ω_t = X(t,Ω₀) also has a C^{1,γ} boundary for all t. This yields Theorem 1.1: vortex‑patch boundaries retain their Hölder regularity under the QGSW dynamics. The proof mirrors the classical Euler results of Chemin and Bertozzi–Constantin but requires new kernel estimates because K is non‑homogeneous.

The second major contribution is the ε→0 limit. The authors show that K_ε converges to the Euler kernel K₀(x)=x⊥/(2π|x|²) in a quantitative way, and that the difference K_ε−K₀ is O(ε) in the appropriate norms. For initial data in the little Hölder space c^γ_c, they compare the QGSW solution q^ε with the Euler solution q^{Euler} by decomposing the error into a term involving the kernel difference and a term involving the difference of flow maps. Using the previously established C^γ bounds and Grönwall’s inequality, they prove Theorem 6.1:
 ‖q^ε(t)−q^{Euler}(t)‖_{C^γ} ≤ C ε,
uniformly on a fixed time interval. This gives a rigorous justification of the formal limit from QGSW to Euler.

Finally, the paper discusses related work, noting that a recent preprint by Tan, Xue, and Xue treats a broader class of active‑scalar equations that includes QGSW, obtaining analogous results with different techniques. Overall, the article provides a comprehensive analysis of vortex‑patch regularity and model convergence for the quasi‑geostrophic shallow‑water equations, extending classical Euler theory to a physically relevant, non‑homogeneous kernel setting.