Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations

Abstract

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^{\infty }$ norm at time $T_{\ast }>0$, while remaining smooth in space and time on ${\mathbb{T}}^d\times ([0,T]\setminus \{T_{\ast }\})$. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier--Stokes flow, which transfers energy from infinitely high to low frequencies. The result holds in all dimensions $d\ge 2$.

minor corrections, 52 pages, 4 figures