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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.02691 |
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| _version_ | 1866908960374652928 |
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| author | Miller, Evan |
| author_facet | Miller, Evan |
| contents | In this paper, we prove a new identity for divergence free vector fields, showing that \begin{equation*} \left<-ΔS,ω\otimesω\right>=0, \end{equation*} where $S_{ij}=\frac{1}{2}\left(\partial_iu_j+\partial_ju_i\right)$ is the symmetric part of the velocity gradient, and $ω=\nabla\times u$ is the vorticity. This identity will allow us to understand the interaction of different aspects of the nonlinearity in the Navier--Stokes equation from the strain and vorticity perspective, particularly as they relate to the depletion of the nonlinearity by advection. We will prove global regularity for the strain-vorticity interaction model equation, a model equation for studying the impact of the vorticity on the evolution of strain which has the same identity for enstrophy growth as the full Navier--Stokes equation. We will also use this identity to obtain several new regularity criteria for the Navier--Stokes equation, one of which will help to clarify the circumstances in which advection can work to deplete the nonlinearity, preventing finite-time blowup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02691 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the interaction of strain and vorticity for solutions of the Navier--Stokes equation Miller, Evan Analysis of PDEs 35Q30 In this paper, we prove a new identity for divergence free vector fields, showing that \begin{equation*} \left<-ΔS,ω\otimesω\right>=0, \end{equation*} where $S_{ij}=\frac{1}{2}\left(\partial_iu_j+\partial_ju_i\right)$ is the symmetric part of the velocity gradient, and $ω=\nabla\times u$ is the vorticity. This identity will allow us to understand the interaction of different aspects of the nonlinearity in the Navier--Stokes equation from the strain and vorticity perspective, particularly as they relate to the depletion of the nonlinearity by advection. We will prove global regularity for the strain-vorticity interaction model equation, a model equation for studying the impact of the vorticity on the evolution of strain which has the same identity for enstrophy growth as the full Navier--Stokes equation. We will also use this identity to obtain several new regularity criteria for the Navier--Stokes equation, one of which will help to clarify the circumstances in which advection can work to deplete the nonlinearity, preventing finite-time blowup. |
| title | On the interaction of strain and vorticity for solutions of the Navier--Stokes equation |
| topic | Analysis of PDEs 35Q30 |
| url | https://arxiv.org/abs/2407.02691 |