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

Abstract: We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on $\mathbb R<sup>2$. More precisely, for every $1&lt;s\&lt;2$, $$T\&gt;0$$ and $\varepsilon&gt;0$, we construct initial data $θ<em>0\in H^s(\mathbb R<sup>2)$ with $|θ_0|</em>{H<sup>s}\leq\varepsilon$ that generate a solution $θ$ on $[0,T]$ such that [ θ\in L^\infty([0,T];H^{1+γ}(\mathbb R^2)), \quad θ(t)\notin H^σ(\mathbb R^2) \quad\text{for every }t\in(0,T],\ σ>σ*(s), ] where $γ&gt;0$ and $σ</em><em>(s)$ satisfy [ 1+γ<σ_(s):= \frac{s(3-s)}{1+2s-s^2}<s. ] The solution is unique in a determined family of classical solutions with initial datum $θ_0$.