Global Regularity of Solutions to the Inviscid Surface Quasi-Geostrophic Equation

Global Regularity of Solutions to the Inviscid Surface Quasi-Geostrophic Equation

Definitions and mathematical statement

For the scalar $\theta(t,x)$ on ${\mathbb{R}}^2$, let $R_j$ be the Riesz transform with Fourier multiplier $-i\xi_j/|\xi|$ and set $R^\perp\theta=(-R_2\theta,R_1\theta)$. Inviscid SQG is \[ \partial_t\theta+u\cdot\nabla\theta=0,\qquad u=R^\perp\theta, \qquad\theta(0,x)=\theta_0(x). \] The problem is global smooth existence for every permitted smooth decaying finite-energy datum, or a finite-time breakdown for one such datum. "Inviscid" means no dissipative fractional-Laplacian term is present. The precise initial norm and smoothness class should follow the cited formulation rather than a weaker low-regularity ill-posedness result.

Formulation and concept references: [S1].

Short English statement

Can a smooth inviscid SQG flow form a singularity in finite time, or must every allowed smooth initial state remain smooth forever?

Research attempt

Current frontier and data class.

The supplied paper's smooth norm-inflation examples remain smooth on their stated interval; its continuous-loss theorem starts in $H^s$, $3/2<s<2$, not in the smooth class of this target. These results cannot be relabeled finite-time blow-up from one smooth finite-energy datum. [S1][E3] Hou--Qin--Sire--Wu's 2026 construction deliberately permits infinite energy and a one-dimensional reduction; its ansatz is not a smooth decaying finite-energy counterexample here. [E1] Nontrivial global smooth solution families are known, but do not establish the every-datum assertion. [E2]

Attack route.

A continuation criterion suggests controlling time-integrated gradient stretching. A frequency-space alternative exploits an exact cancellation when interacting frequencies have equal length. We derive that cancellation, quantify its gain in a thin annulus, and actively test whether the annulus is preserved. The resulting leakage shows why the initial estimate alone cannot close a global bootstrap. The deductions below are explicit calculations, not a claim of a new regularity theorem.

Attempt: expose the continuation quantity.

For a smooth decaying solution and integer $s\ge3$, let $X=\|\theta\|_{H^s}$, $M=\|\nabla\theta\|_\infty$, and $\Theta_2=\|\theta_0\|_2$. Transport by the divergence-free velocity conserves $\Theta_2$. A dyadic decomposition gives, for $J\ge1$, \[ \|\nabla u\|_\infty \le C\Theta_2+CJM+C_s2^{-(s-2)J}X. \] Indeed low frequencies are bounded using the $L^2$ norm; each middle block is a localized order-zero multiplier applied to $\nabla\theta$, with uniformly bounded convolution $L^1$ kernel. At high frequencies, Bernstein in two dimensions gives $\|\nabla u_j\|_\infty\le C2^{2j}\|\theta_j\|_2$, and Cauchy--Schwarz sums the geometric tail using $s>2$. Choosing $J=\lceil\log_2(e+X)/(s-2)\rceil$ yields \begin{equation}\label{wsref-0-33-30-30-30-36-37-35-33-3a-6c-6f-67} \|\nabla u\|_\infty \le C_s\left(1+\Theta_2+M\log(e+X)\right). \end{equation}

The differentiated transport energy estimate is \[ X'\le C_s\left(\|\nabla u\|_\infty+M\right)X. \] Here the leading transport term cancels by $\nabla\cdot u=0$; the Leibniz commutator terms are bounded by $C_s(\|\nabla u\|_\infty\|\theta\|_{H^s} +\|u\|_{H^s}\|\nabla\theta\|_\infty)$, using product interpolation for intermediate derivative orders. The Riesz multiplier is bounded on $H^s$, so $\|u\|_{H^s}\le CX$. With $W=\log(e+X)\ge1$, (30006753:log) therefore implies \[ W'\le C_s(1+\Theta_2+M)W. \] Consequently $\int_0^TM(t)\,dt<\infty$ bounds every initially finite $H^s$ norm and gives continuation by the usual local smooth theory. This proves the familiar sufficient criterion rather than deriving its unknown hypothesis from conserved scalar norms.

For $g=\nabla\theta\ne0$ and $\xi=g/|g|$, the exact material identity is \[ (\partial_t+u\cdot\nabla)g=-(Du)^{\mathsf T}g,\qquad (\partial_t+u\cdot\nabla)\log|g| =-\xi^{\mathsf T}(Du)^{\mathsf T}\xi. \] There is no pointwise sign in this stretching term. Controlling its positive accumulation along trajectories is a possible geometric route, but incompressibility alone supplies no such bound.

A thin-shell cancellation estimate.

Use $\widehat\theta(\xi)=\int e^{-ix\cdot\xi}\theta(x)\,dx$ and $\xi^\perp=(-\xi_2,\xi_1)$. The exact velocity convention in the question gives $\widehat u(\xi)=-i\xi^\perp|\xi|^{-1}\widehat\theta(\xi)$. Symmetrizing the Fourier convolution yields \begin{equation}\label{wsref-0-33-30-30-30-36-37-35-33-3a-6b-65-72-6e-65-6c} \widehat{u\cdot\nabla\theta}(q) =\frac{1}{2(2\pi)^2}\int_{k+\ell=q} (k^\perp\cdot\ell) \left(\frac1{|k|}-\frac1{|\ell|}\right) \widehat\theta(k)\widehat\theta(\ell)\,dk. \end{equation} The factor vanishes exactly when the two radii agree, as well as for parallel interacting frequencies. Suppose at a single time \[ \operatorname{supp}\widehat\theta\subset \{R\le|\xi|\le(1+\epsilon)R\}, \qquad R\ge1,\quad0<\epsilon\le\tfrac12. \] The symmetrized multiplier has magnitude at most $C\epsilon R$, and its output support has radius at most $3R$. Young's convolution inequality and Cauchy--Schwarz on the annulus then prove \begin{align} \|u\cdot\nabla\theta\|_{H^s} &\le C_s\epsilon R(1+R)^s \|\widehat\theta\|_1\|\theta\|_2\notag\\ &\le C_s\epsilon^{3/2}R^2(1+R)^s\|\theta\|_2^2. \label{wsref-0-33-30-30-30-36-37-35-33-3a-73-68-65-6c-6c} \end{align} The second step uses annulus area $O(\epsilon R^2)$. This is a proved instantaneous estimate, not an assumption of random phases or a conservation law for the spectral support.

Stress test: the annulus leaks immediately.

Take $0<\epsilon<1/4$ and frequencies \[ k=R(1+\epsilon/4,0),\qquad \ell=R(0,1+3\epsilon/4). \] They lie strictly within the annulus, but $|k+\ell|>(1+\epsilon)R$. The symmetrized multiplier in (30006753:kernel), including its factor $1/2$ but excluding $(2\pi)^{-2}$, is $\epsilon R/4\ne0$. Thus orthogonal frequencies of slightly different radii generate a frequency outside the initial annulus.

This can be realized with smooth finite-energy data, not just plane waves. Choose nonnegative smooth Fourier bumps with small disjoint supports around $\pm k,\pm\ell$, symmetric under $\xi\mapsto-\xi$. Their inverse transform is real and Schwartz. At frequencies near $k+\ell$, only the two cross interactions contribute; their symmetrized multiplier has the same strict sign on sufficiently small supports. The convolution is consequently nonzero on an open output set outside the annulus. The local smooth solution is differentiable in a Sobolev space, so projection to that output set gives $\widehat\theta(t)=t\,\partial_t\widehat\theta(0)+o(t)$ there in $L^2$. Its spectral support has left the annulus for sufficiently small positive time. Equation (30006753:shell) cannot therefore be iterated by assuming the same support restriction.

Exact support on one circle is not a nontrivial finite-energy escape from this difficulty: an $L^2({\mathbb{R}}^2)$ Fourier transform supported on a measure-zero circle is zero. On a torus monochromatic modes may be nontrivial and stationary, but changing domain would change the target. Radial Schwartz data on ${\mathbb{R}}^2$ do give stationary solutions, since $u=-\nabla^\perp\Lambda^{-1}\theta$ is tangent to circles and $\nabla\theta$ is radial. That special case does not control angular perturbations or arbitrary data.

Finally $\theta_\lambda(t,x)=\theta(\lambda t,\lambda x)$ is the amplitude-preserving scaling. Initially it keeps $\|\theta\|_\infty$ fixed, decreases $\|\theta\|_2$ by $\lambda^{-1}$, and increases $\|\nabla\theta\|_\infty$ by $\lambda$. Bounded conserved upper norms therefore cannot give an a priori gradient bound without additional structure. This is a scaling countertest, not a singular solution.

Outcome.

Outcome: Exploratory approach with unresolved gap. The attempt supplies the continuation calculation, an exact equal-radius cancellation and a quantitative thin-annulus bound. It also proves that smooth finite-energy data can immediately generate frequencies outside that annulus, invalidating the proposed invariant-shell bootstrap. The missing step is control of the newly generated spectrum or of integrated gradient stretching for arbitrary smooth data. Neither a global bound nor a finite-time singularity in the target class is established.

Sources

• S1 Cordoba, Martinez-Zoroa, and Ozanski, Instantaneous Continuous Loss of Regularity for the SQG Equation, Introduction. https://www.sciencedirect.com/science/article/pii/S0001870825004517. Catalogue formulation source.
• E1 T. Y. Hou, X. Qin, Y. Sire and Y. Wu, Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy, arXiv:2603.11301v1, 11 March 2026, Introduction and Theorem 1.1. https://arxiv.org/html/2603.11301v1.
• E2 A. Castro, D. Córdoba and J. Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, Memoirs of the AMS 266 $2020$, no. 1292; arXiv:1603.03325, abstract and solution-family theorem. https://arxiv.org/abs/1603.03325.
• E3 D. Córdoba, L. Martínez-Zoroa and W. S. Ożański, Instantaneous continuous loss of regularity for the SQG equation, accessible manuscript arXiv:2409.18900v1, Section 1.1, Theorem 1 and continuous-loss statement; manuscript of [S1]. https://arxiv.org/html/2409.18900v1.