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

Alexey Cheskidov, Mimi Dai, Stan Palasek

TL;DR

This work reveals a new instantaneous Type I blow-up phenomenon for the Navier–Stokes equations: starting from smooth initial data, there exists a smooth evolution that exhibits right-infinite growth of the $L^{\infty}$ norm immediately after a blow-up time $T_*$, driven by an inverse energy cascade from infinite wavenumber. The authors construct a leading-order multiscale velocity field $v$ via an infinite hierarchy of oscillatory building blocks and couple it with a small corrector $w$ to produce a genuine NSE solution; crucially, the energy flux $\Pi_N([T_*,t])$ becomes unbounded as $N\to\infty$ at $T_*$, enabling a bifurcation into infinitely many smooth evolutions with the same initial data and thus non-uniqueness. The mechanism operates in all dimensions $d\ge 2$ and places the constructed solutions on the sharp boundary of known regularity criteria, linking instantaneous blow-up to a nonlinear, fully nonlinear energy cascade. The results have potential implications for our understanding of well-posedness and turbulence-like energy transfer in viscous incompressible flows, showing that even smooth flows can spontaneously lose uniqueness through internal spectral dynamics.

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 $d\geq2$.

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

TL;DR

This work reveals a new instantaneous Type I blow-up phenomenon for the Navier–Stokes equations: starting from smooth initial data, there exists a smooth evolution that exhibits right-infinite growth of the norm immediately after a blow-up time , driven by an inverse energy cascade from infinite wavenumber. The authors construct a leading-order multiscale velocity field via an infinite hierarchy of oscillatory building blocks and couple it with a small corrector to produce a genuine NSE solution; crucially, the energy flux becomes unbounded as at , enabling a bifurcation into infinitely many smooth evolutions with the same initial data and thus non-uniqueness. The mechanism operates in all dimensions and places the constructed solutions on the sharp boundary of known regularity criteria, linking instantaneous blow-up to a nonlinear, fully nonlinear energy cascade. The results have potential implications for our understanding of well-posedness and turbulence-like energy transfer in viscous incompressible flows, showing that even smooth flows can spontaneously lose uniqueness through internal spectral dynamics.

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 norm at time , while remaining smooth in space and time on . 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 .

Paper Structure

This paper contains 26 sections, 17 theorems, 264 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Finite-time vs. instantaneous blow-up
  4. Criticality and Type I blow-up
  5. Previous non-uniqueness constructions
  6. Convex integration
  7. Self-similar solutions
  8. Non-uniqueness based on dyadic scenarios
  9. Sharpness of the non-uniqueness theorem
  10. Inverse energy cascade
  11. Main ideas of the construction
  12. Preliminaries
  13. Notation
  14. Koch--Tataru spaces
  15. Weak solutions
  16. ...and 11 more sections

Key Result

Theorem 1.1

For any $d\geq2$ and $u_0 \in C^\infty(\mathbb{T}^d)$ divergence-free, there exists $T>0$ such that the following holds. For any $T_*\in[0,T)$, there exists a weak solution $u(t)$ of eq:NSE on $\mathbb T^d\times[0,T]$ such that:

Figures (4)

  • Figure 1: Blow-up from the right.
  • Figure 2: Continuous family of solutions $u^{(\sigma)}$with the same initial data.
  • Figure 3: Schematic depiction of possible scenarios following a finite-time blow-up: (A) Formation of a spatial singularity at $t=T_*$; (B) Absence of a spatial singularity at $t=T_*$ with instantaneous blow-up from the right; (C) Absence of spatial singularity at $t=T_*$ with a classical solution starting from $t=T_*$.
  • Figure 4: Time evolution of $\|u_k(t)\|_{L^\infty}$ and $\|u_{k+1}(t)\|_{L^\infty}$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 28 more