Instantaneous continuous loss of regularity for the SQG equation

Given $s\in (3\mathrm{/}2,2)$ and $\epsilon >0$, we construct a compactly supported initial data ${\theta }_0$ such that $\mathrm{\parallel }{\theta }_0{\mathrm{\parallel }}_{H^s}\le \epsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $\theta$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on ${\mathbb{R}}^2\times [0,T]$, and, for each $t\in [0,T]$, $\theta (\cdot ,t)\in H^{s\mathrm{/}(1+ct)}$ and $\theta (\cdot ,t)\in ̸H^{\beta }$ for any $\beta >s\mathrm{/}(1+ct)$. Moreover, $\theta$ is unique among all solutions with initial condition ${\theta }_0$ which belong to $C([0,T];H^{1+\alpha })$ for any $\alpha >0$ and is continuous and differentiable in $t$ and in $x$ on ${\mathbb{R}}^2\times [0,T]$.