Instantaneous continuous loss of regularity for the SQG equation

Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $\theta_0$ such that $\| \theta_0 \|_{H^s}\leq \varepsilon$ 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 $t\in [0,T]$, $\theta (\cdot ,t ) \in {H^{s/(1+ct)}}$ and $\theta (\cdot ,t ) \not \in {H^\beta }$ for any $\beta > s/(1+ct)$. 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 $\mathbb{R}^2\times [0,T]$.