Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities
Dongho Chae, In-Jee Jeong, Sung-Jin Oh
TL;DR
The paper addresses the strong illposedness of the generalized SQG equation with singular multipliers by revealing a degenerate-dispersion mechanism. It develops a general framework for degenerating wave packets, rooted in a Hamilton-Jacobi phase and a transport amplitude, and proves linear and nonlinear illposedness in Sobolev spaces, including dissipative variants, with sharp growth rates tied to the symbol γ. The construction hinges on precise PDO calculus, oscillatory integral estimates, and a generalized energy identity to enable a duality-based illposedness argument. The results illuminate the delicate balance between dispersion, singular velocities, and dissipation, and pinpoint the regimes where well-posedness may be recovered or fails, offering a pathway to understanding related 2D geophysical models and their mathematical limits.
Abstract
We prove strong nonlinear illposedness results for the generalized SQG equation $$\partial_t θ+ \nabla^\perp Γ[θ] \cdot \nabla θ= 0 $$ in any sufficiently regular Sobolev spaces, when $Γ$ is a singular in the sense that its symbol satisfies $|Γ(ξ)|\to\infty$ as $|ξ|\to\infty$ with some mild regularity assumptions. The key mechanism is degenerate dispersion, i.e., the rapid growth of frequencies of solutions around certain shear states, and 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 $Γ$. Our illposedness results are completely sharp in view of various existing wellposedness statements as well as those from our companion paper. Key to our proofs is a novel construction of degenerating wave packets for the class of linear equations $$\partial_t φ+ ip(t,X,D)φ= 0$$ where $p(t,X,D)$ is a pseudo-differential operator which is self-adjoint in $L^2$, degenerate, and dispersive. Degenerating wave packets are approximate solutions to the above linear equation with spatial and frequency support localized at $(X(t),Ξ(t))$, which are solutions to the bicharacteristic ODE system associated with $p(t,x,ξ)$. These wave packets explicitly show degeneration as $X(t)$ approaches a point where $p$ vanishes, which in particular allows us to prove illposedness in topologies finer than $L^2$. While the equation for the wave packet can be formally obtained from a Taylor expansion of the symbol near $ξ=Ξ(t)$, the difficult part is to rigorously control the error in sufficiently long timescales, which is obtained by sharp estimates for not only degenerating wave packets but also for oscillatory integrals which naturally appear in the error estimate.