Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation

We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on ${\mathbb{R}}^2$. More precisely, for every $1<s<2$, $T>0$ and $\epsilon >0$, we construct initial data ${\theta }_0\in H^s({\mathbb{R}}^2)$ with $\mathrm{\parallel }{\theta }_0{\mathrm{\parallel }}_{H^s}\le \epsilon$ that generate a solution $\theta$ on $[0,T]$ such that $$\theta \in L^{\mathrm{\infty }}([0,T];H^{1+\gamma }({\mathbb{R}}^2)),\theta (t)\mathrm{\notin }{H}^{\sigma }({\mathbb{R}}^2)\text{for every }t\in (0,T],\sigma >{\sigma }_{\ast }(s),$$ where $\gamma >0$ and ${\sigma }_{\ast }(s)$ satisfy $$1+\gamma <{\sigma }_{\ast }(s):=\frac{s(3-s)}{1+2s-s^2}<s\mathrm{.}$$ The solution is unique in a determined family of classical solutions with initial datum ${\theta }_0$.