Small scale formation for the 2D Boussinesq equation
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909430930472960 |
|---|---|
| author | Kiselev, Alexander Park, Jaemin Yao, Yao |
| author_facet | Kiselev, Alexander Park, Jaemin Yao, Yao |
| contents | We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^2}\gtrsim t^α$ for some $α>0$. For the inviscid equation in the strip, we construct examples satisfying $\|ω(t)\|_{L^\infty}\gtrsim t^3$ and $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^\infty} \gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_05070 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Small scale formation for the 2D Boussinesq equation Kiselev, Alexander Park, Jaemin Yao, Yao Analysis of PDEs 35B40, 35Q31, 35Q35 We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^2}\gtrsim t^α$ for some $α>0$. For the inviscid equation in the strip, we construct examples satisfying $\|ω(t)\|_{L^\infty}\gtrsim t^3$ and $\sup_{τ\in[0,t]} \|\nabla ρ(τ)\|_{L^\infty} \gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth. |
| title | Small scale formation for the 2D Boussinesq equation |
| topic | Analysis of PDEs 35B40, 35Q31, 35Q35 |
| url | https://arxiv.org/abs/2211.05070 |