Illposedness of incompressible fluids in supercritical Sobolev spaces

We prove that the 3D Euler and Navier-Stokes equations are strongly illposed in supercritical Sobolev spaces. In the inviscid case, for any 0<s< 52, we construct a Cᶜ initial velocity field with arbitrarily small H^s norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large Ḣ^s norm inflation almost instantaneously. In the viscous case, the same Ḣ^s norm inflation occurs in the 3D Navier-Stokes equations for 0<s< 12, where s = 12 is scaling critical for this equation.