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

TL;DR

This paper constructs smooth solutions to the Navier-Stokes equations that blow up instantaneously and are non-unique, demonstrating fundamental issues in the equations' well-posedness.

Contribution

It introduces the first known example of instantaneous Type I blow-up and non-uniqueness of smooth solutions for Navier-Stokes equations, including a classical inverse energy cascade.

Findings

Existence of solutions with instantaneous Type I blow-up.

Demonstration of non-uniqueness of smooth solutions.

Construction of a classical inverse energy cascade in Navier-Stokes flows.

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 $T_*>0$, while remaining smooth in space and time on $\mathbb T^d\times([0,T]\setminus\{T_*\})$. 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 $d\geq2$.