Instantaneous continuous loss of regularity for the SQG equation

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