Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.09556 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917195549769728 |
|---|---|
| author | Cheskidov, Alexey Dai, Mimi Palasek, Stan |
| author_facet | Cheskidov, Alexey Dai, Mimi Palasek, Stan |
| contents | 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 $d\geq2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations Cheskidov, Alexey Dai, Mimi Palasek, Stan Analysis of PDEs 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 $d\geq2$. |
| title | Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.09556 |