Illposedness via Degenerate Dispersion for Generalized Surface Quasi-Geostrophic Equations with Singular Velocities

We prove strong nonlinear illposedness results for the generalized SQG equation ∂ t θ + ∇ ⊥ Γ [ θ ] ⋅ ∇ θ = 0 \begin{equation*} \begin {split} \partial _t \theta + \nabla ^\perp \Gamma [\theta ] \cdot \nabla \theta = 0 \end{split} \end{equation*} in any sufficiently regular Sobolev spaces, when Γ \Gamma is a singular multiplier in the sense that its symbol satisfies | Γ ( ξ ) | → ∞ |\Gamma (\xi )|\rightarrow \infty as | ξ | → ∞ |\xi |\rightarrow \infty together some mild regularity assumptions. The key mechanism is degenerate dispersion, i.e., the rapid growth of frequencies of solutions around certain shear states, as in the second and third author’s earlier work on Hall-magnetohydrodynamics [In-Jee Jeong and Sung-Jin Oh, On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity I: Illposedness near degenerate stationary solutions , Ann. PDE 8 (2022), no. 2, Paper No. 15, 106]. The robustness of our method allows one to extend linear and nonlinear illposedness to fractionally dissipative systems, as long as the order of dissipation is lower than that of Γ \Gamma . Our illposedness results are completely sharp in view of various existing wellposedness statements as well as those from our companion paper [Dongho Chae, In-Jee Jeong, Jungkyoung Na, and Sung-Jin Oh, Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations , Comm. Math. Phys. 406 (2025), no. 4, Paper No. 75]. Key to our proofs is a novel construction of degenerating wave packets for the class of linear equations ∂ t ϕ