Illposedness of incompressible fluids in supercritical Sobolev spaces

Source abstract

We prove that the 3-dimensional (3D) Euler and Navier–Stokes equations are strongly ill posed in supercritical Sobolev spaces. In the inviscid case, for any 0<s<5 2, we construct a Cc∞ initial velocity field with arbitrarily small Hs norm for which the unique local-in-time smooth solution of the 3D Euler equation develops large H˙s norm inflation almost instantaneously. In the viscous case, the same H˙s norm inflation occurs in the 3D Navier–Stokes equation for 0<s<1 2, where s=1 2 is scaling critical for this equation.