Math Conjectures

Global regularity for inviscid surface quasi-geostrophic flow

OPENLandmarkOpen problemProposed 1994 · Canonical special case

Canonical statement

For every θ0∈S(R2)\theta_0\in\mathcal S(\mathbb R^2), the unique maximal classical solution of
∂tθ+u⋅∇θ=0,u=∇⊥(−Δ)−1/2θ=(−R2θ,R1θ),θ(⋅,0)=θ0 \partial_t\theta+u\cdot\nabla\theta=0,\qquad u=\nabla^\perp(-\Delta)^{-1/2}\theta =(-R_2\theta,R_1\theta),\qquad \theta(\cdot,0)=\theta_0
exists for every t≥0t\ge0 and is smooth, where RjR_j is the jj-th Riesz transform.
View source LaTeX
For every \(\theta_0\in\mathcal S(\mathbb R^2)\), the unique maximal classical solution of \[ \partial_t\theta+u\cdot\nabla\theta=0,\qquad u=\nabla^\perp(-\Delta)^{-1/2}\theta =(-R_2\theta,R_1\theta),\qquad \theta(\cdot,0)=\theta_0 \] exists for every \(t\ge0\) and is smooth, where \(R_j\) is the \(j\)-th Riesz transform.
PDE-003:[ConstantinMajdaTabak1994SQG][Cordoba1998SQGScenario][Cameron2024GeometricSQG]
Partial differential equationsMathematical physics

Notes

The problem concerns the inviscid surface quasi-geostrophic (SQG) equation, in which a scalar θ\theta on R2\mathbb R^2 is transported by the velocity u=∇⊥(−Δ)−1/2θu=\nabla^\perp(-\Delta)^{-1/2}\theta built from its Riesz transforms. It asks whether every Schwartz initial datum yields a classical solution that remains smooth for all time. The question was posed in 1994 by Constantin, Majda, and Tabak, who introduced the equation as a two-dimensional model whose front-formation and stretching mechanisms parallel those of the three-dimensional Euler equations [ConstantinMajdaTabak1994SQG].

Early candidate singularity scenarios were ruled out: Córdoba proved that the simple hyperbolic-saddle closing of level sets cannot produce blow-up [Cordoba1998SQGScenario], and geometric methods have since been developed for the generalized SQG family [Cameron2024GeometricSQG]. Global regularity is known for the critically dissipative SQG equation, but those arguments use the dissipation and do not apply to the inviscid equation.

At present there is neither a smooth finite-time singularity nor a general a priori regularity bound for inviscid SQG, and the problem is open in both directions.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.