Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in R^3. Namely, for any s (0, 3/2) and >0, we construct a divergence-free initial vorticity ω₀ defined in R^3 satisfying \| ω₀ \|₇⌁, as well as T>0, c>0 and a corresponding local-in-time solution ω such that, for each t 0, T, ω (, t) H^{s-ct{1+ct}} and ω (, t) H^β for any β> s-ct1+ct. Moreover, ω is unique among all solutions with initial condition ω₀ which are locally C² and belong to C (0, T;Lᵖ) for any p>3.