Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

Abstract

We construct solutions of the 2D incompressible Euler equations in ${\mathbb{R}}^2\times [0,\mathrm{\infty })$ such that initially the velocity is in the super-critical Sobolev space $H^{\mathit{\beta }}$ for $1<\mathit{\beta }<2$, but is not in $H^{{\mathit{\beta }}^{\prime }}$ for ${\mathit{\beta }}^{\prime }>1+\frac{(3-\mathit{\beta })(\mathit{\beta }-1)}{2-{(\mathit{\beta }-1)}^2}$ for any $t\in (0,\mathrm{\infty })$. These solutions are not in the Yudovich class, but they exist globally in time and they are unique in a determined family of classical solutions.