Flexibility for the SQG Equation with an $L^{4/3+}$ Active Scalar

Abstract: We develop a new convex-integration scheme, inspired by \cite{BCK26}, for the inviscid surface quasi-geostrophic equation on the two-dimensional torus. For the explicit, nonoptimized exponent $\bar p=\frac{4}{3}+10<sup>{-5},$ we prove a flexibility theorem for weak solutions in the standard momentum formulation with active scalar [ θ\in C([0,1];L^{\bar p}(\mathbb T^2)). ] More precisely, any two prescribed mean-zero states in $L^{\bar p}(\mathbb T<sup>2)$ can be approximated at the initial and final times by such a solution. The perturbations are constructed from localized, concentrated traveling SQG profiles whose centers move along rational directions and whose radii depend on the Reynolds stress. Time averages of auxiliary sources along these trajectories reconstruct the preceding-stage stress, while a two-dimensional bilinear null-form estimate compensates for the derivative loss caused by the nonlocal constitutive law. Exploiting the time-locality of the iteration, we also obtain a dense subset of the mean-zero space $L^{\bar p}(\mathbb T<sup>2)$ such that every initial datum in this subset admits at least two distinct momentum weak solutions. Thus, the construction establishes both flexibility and nonuniqueness beyond the concentration-critical exponent $p=4/3$.

• The paper develops a convex-integration scheme proving flexibility for inviscid SQG in C([0,1];L^{4/3+10^{-5}}), allowing arbitrary mean-zero endpoint states to be approximated by weak solutions.
• The construction combines traveling counter-rotating dipoles, time-averaged Reynolds-stress reconstruction, variable-radius profiles, and a bilinear null-form estimate that offsets the derivative loss from the nonlocal velocity law.
• The results yield dense nonuniqueness and weak solutions that fail to conserve the SQG Hamiltonian, while leaving the broader range 4/3 ≤ p < 3 and the endpoint p = 4/3 open.

This paper by Brué, Jin, and Nguyen (2608.16641) develops a convex-integration scheme for the inviscid surface quasi-geostrophic (SQG) equation on the two-dimensional torus and proves two theorems concerning weak solutions whose active scalar has integrability just above the critical threshold $p=4/3$. The work extends the moving-dipole construction of Brué, Colombo, and Kumar for two-dimensional Euler to the SQG momentum formulation, where the transport velocity is one derivative more singular than the potential velocity.

Main results

The paper works with the SQG equation $\partial_t\theta+u\cdot\nabla\theta=0$, $u=\nabla^\perp\Lambda^{-1}\theta$, on $\mathbb T^2$, interpreted through the momentum formulation for the potential velocity $v=\nabla^\perp(-\Delta)^{-1}\theta$:

$$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$

Weak solutions are defined via the commutator form of the nonlinearity, which remains meaningful for $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$. The Sobolev embedding $W^{1,p}\hookrightarrow H^{1/2}$ holds precisely when $p\ge 4/3$, which identifies $L^{4/3}$ as the natural Lebesgue threshold for this solution class; this is the endpoint treated by De Rosa, Latocca, and Park via vanishing-viscosity limits.

For the explicit, non-optimized exponent $\partial_t\theta+u\cdot\nabla\theta=0$0, the first main theorem establishes flexibility: any two prescribed mean-zero states in $\partial_t\theta+u\cdot\nabla\theta=0$1 can be approximated in $\partial_t\theta+u\cdot\nabla\theta=0$2 at times $\partial_t\theta+u\cdot\nabla\theta=0$3 and $\partial_t\theta+u\cdot\nabla\theta=0$4 by a weak solution $\partial_t\theta+u\cdot\nabla\theta=0$5. The second theorem uses the time-locality of the iteration to produce a dense subset $\partial_t\theta+u\cdot\nabla\theta=0$6 of the mean-zero subspace of $\partial_t\theta+u\cdot\nabla\theta=0$7 such that every datum in $\partial_t\theta+u\cdot\nabla\theta=0$8 admits at least two distinct momentum weak solutions. The authors state this is the first nonuniqueness result for SQG in $\partial_t\theta+u\cdot\nabla\theta=0$9 with $u=\nabla^\perp\Lambda^{-1}\theta$0 above the critical threshold.

An immediate consequence of the flexibility theorem concerns the Hamiltonian $u=\nabla^\perp\Lambda^{-1}\theta$1. Choosing endpoint states with different Hamiltonians yields a weak solution that does not conserve $u=\nabla^\perp\Lambda^{-1}\theta$2, since $u=\nabla^\perp\Lambda^{-1}\theta$3 is continuous in the $u=\nabla^\perp\Lambda^{-1}\theta$4 topology. This is a strong claim: it is the first failure of Hamiltonian conservation at any $u=\nabla^\perp\Lambda^{-1}\theta$5, and the resulting solutions cannot arise as vanishing-viscosity limits of the type constructed by De Rosa, Latocca, and Park, whose procedures produce Hamiltonian-conserving solutions. On the rigidity side, $u=\nabla^\perp\Lambda^{-1}\theta$6 guarantees conservation, so the gap between $u=\nabla^\perp\Lambda^{-1}\theta$7 and $u=\nabla^\perp\Lambda^{-1}\theta$8 remains.

Traveling SQG building blocks

The seed profile is the compactly supported Lipschitz traveling counter-rotating circular pair of Cao, Qin, Zhan, and Zou, satisfying $u=\nabla^\perp\Lambda^{-1}\theta$9 with $\mathbb T^2$0 and zero total mass. After rotation and rescaling at scale $\mathbb T^2$1, the potential velocity scales as $\mathbb T^2$2 while the scalar scales as $\mathbb T^2$3, so that $\mathbb T^2$4 and $\mathbb T^2$5 are invariant. The nonzero first moment $\mathbb T^2$6 of the profile produces the key identity

$\mathbb T^2$7

which forces the exponent $\mathbb T^2$8 in the radius-amplitude relation used in the iteration.

Two structural obstacles are handled at this stage. First, the whole-space potential velocity has a noncompact far field; subtracting the exterior harmonic gradient and periodizing, then restoring incompressibility with a periodic gradient corrector, yields a localized divergence-free block $\mathbb T^2$9 with $v=\nabla^\perp(-\Delta)^{-1}\theta$0. Second, the block must satisfy two exact identities — the constant-speed momentum equation with a small symmetric error $v=\nabla^\perp(-\Delta)^{-1}\theta$1, and the $v=\nabla^\perp(-\Delta)^{-1}\theta$2-derivative identity $v=\nabla^\perp(-\Delta)^{-1}\theta$3 — which allow the center, amplitude, and radius of the block to vary in time. A fixed-scale spatial mollification (independent of $v=\nabla^\perp(-\Delta)^{-1}\theta$4, so that $v=\nabla^\perp(-\Delta)^{-1}\theta$5 commutes with convolution) makes the profile family smooth jointly in $v=\nabla^\perp(-\Delta)^{-1}\theta$6 while preserving the identities and estimates. The gradient corrector acts as a pressure gauge: since it is a gradient, its contribution to the second slot of the momentum nonlinearity $v=\nabla^\perp(-\Delta)^{-1}\theta$7 vanishes identically.

The convex-integration iteration

At stage $v=\nabla^\perp(-\Delta)^{-1}\theta$8, the mollified Reynolds stress is decomposed into four positive rank-one components $v=\nabla^\perp(-\Delta)^{-1}\theta$9 along rational directions with long periods $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$0. The principal perturbation consists of a single traveling block at any given time: the coarse time interval of length $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$1 is partitioned into four disjoint active subintervals, so distinct principal blocks never interact. The spatially varying radius is chosen by $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$2, where $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$3 is the intervalwise time average of $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$4, and the center moves along the rational direction $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$5 at speed inversely proportional to $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$6. Consequently, the block completes at least $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$7 closed periods per active subinterval, and the time spent in a region is weighted by $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$8.

The stress is reconstructed through temporal averaging rather than low-frequency wave interactions: the time average of an auxiliary vector source along the trajectories reproduces $$\partial_t v+u\cdot\nabla v-(\nabla v)^Tu+\nabla p=0,\qquad \operatorname{div}v=0,\qquad u=\Lambda v.$$9 up to an error of size $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$0. An auxiliary fixed-width profile with an explicitly uniform orbit average replaces the variable-radius source, and a Leray-projected time corrector absorbs the zero-mean temporal oscillation, vanishing at the endpoints of each coarse interval. The new Reynolds stress comprises six components — linear interaction, corrector interaction, time-freezing, auxiliary-average, building-block, and source-replacement errors — each bounded by $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$1 under the parameter hierarchy.

The null form and why the gain is small

The central new difficulty relative to the Euler construction of [BCK26] is the nonlocal relation $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$2: the linear interaction with the background contains one derivative of the concentrated profile, and a direct estimate incurs a negative power of the concentration radius. The paper circumvents this with the two-dimensional identity $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$3 and a bilinear null-form estimate

$v\in L^2_{\mathrm{loc}}(I;H^{1/2})$4

proved in the appendix via the multilinear Christ–Journé commutator estimates of Seeger, Smart, and Street. The null form replaces the unfavorable derivative norm of the concentrated perturbation by a norm of its potential velocity, and is essential for closing the linear error.

The null form, however, does not by itself explain the smallness of $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$5. For $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$6 the concentration exponent $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$7 is positive, and the differentiated block satisfies $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$8. Fitting the $v\in L^2_{\mathrm{loc}}(I;H^{1/2})$9 bookkeeping requires $W^{1,p}\hookrightarrow H^{1/2}$0, which for the admissible parameters ($W^{1,p}\hookrightarrow H^{1/2}$1, $W^{1,p}\hookrightarrow H^{1/2}$2, $W^{1,p}\hookrightarrow H^{1/2}$3, $W^{1,p}\hookrightarrow H^{1/2}$4, $W^{1,p}\hookrightarrow H^{1/2}$5, $W^{1,p}\hookrightarrow H^{1/2}$6) forces $W^{1,p}\hookrightarrow H^{1/2}$7, i.e., $W^{1,p}\hookrightarrow H^{1/2}$8. Thus the tiny gain above the endpoint is dictated by the amount of profile concentration the remaining parameter hierarchy can absorb, and the exponent is explicitly non-optimized.

Limitations and open problems

The paper is explicit about the boundaries of the construction. The exponent $W^{1,p}\hookrightarrow H^{1/2}$9 is far from the rigidity threshold $p\ge 4/3$0, and extending flexibility to the full range $p\ge 4/3$1 — in particular constructing Hamiltonian-nonconserving solutions with integrability arbitrarily close to $p\ge 4/3$2 — is left open. For the generalized SQG family $p\ge 4/3$3, the moment scaling, critical exponent, and derivative loss all depend on $p\ge 4/3$4, and it is unknown whether the temporal stress reconstruction and a fractional analogue of the null form survive the modified scaling; even a fixed small dissipation produces a high-frequency error not absorbed by the present iteration, with a frequency-dependent vanishing-dissipation regime suggested as a more accessible first problem. Finally, the whole-space problem is obstructed by the averaging mechanism itself: the construction relies on closed rational trajectories that repeatedly sample the coefficient being reconstructed, whereas on $p\ge 4/3$5 a block escapes, so an extension would require finite sweeping trajectories or a large-box limit with uniform control of moments and nonlocal tails.

Conclusion

The paper establishes flexibility between arbitrary prescribed endpoint states, failure of Hamiltonian conservation, and dense nonuniqueness for inviscid SQG weak solutions with active scalar in $p\ge 4/3$6 for $p\ge 4/3$7, crossing the concentration-critical Lebesgue exponent in the standard momentum formulation. The construction combines traveling SQG profiles with time-dependent radius and position, orbit-averaged temporal reconstruction of the Reynolds stress, and a bilinear null-form estimate that compensates for the derivative loss inherent in the nonlocal constitutive law $p\ge 4/3$8. The quantitative gap between the achieved exponent and both the endpoint $p\ge 4/3$9 and the rigidity threshold $L^{4/3}$0 delineates the current reach of the method.