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 $\bar p=\frac{4}{3}+10^{-5},$ 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 $L^{\bar p}(\mathbb T^2)$ 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 $L^{\bar p}(\mathbb T^2)$ 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 $p=4/3$.