A blow up solution of the Navier-Stokes equations with a critical force
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909443242852352 |
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| author | Zhang, Qi S. |
| author_facet | Zhang, Qi S. |
| contents | A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip
boundary condition is constructed. The scaling factor of the forcing term is the
critical order $-2$. The velocity, which is smooth until its final blow up moment,
is in the energy space through out. Since most physical forces from a point source
in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this
result indicates possible singularity formation under these kind of forces. The result
even holds for some log subcritical forces or some forces in the standard critical space $L^\infty_t
L^{3/2}_x$, including the explicit force: $F=- δ\frac{e^{-|x|^2}}{(|x|^2 + T-t) \,[1+ | \ln (|x|^2 + T-t)|]} (1, 0, 0) $ for any small $δ>0$. The result can also be considered as a step in Scheffer's plan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13896 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A blow up solution of the Navier-Stokes equations with a critical force Zhang, Qi S. Analysis of PDEs 35Q30, 76N10 A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up moment, is in the energy space through out. Since most physical forces from a point source in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this result indicates possible singularity formation under these kind of forces. The result even holds for some log subcritical forces or some forces in the standard critical space $L^\infty_t L^{3/2}_x$, including the explicit force: $F=- δ\frac{e^{-|x|^2}}{(|x|^2 + T-t) \,[1+ | \ln (|x|^2 + T-t)|]} (1, 0, 0) $ for any small $δ>0$. The result can also be considered as a step in Scheffer's plan. |
| title | A blow up solution of the Navier-Stokes equations with a critical force |
| topic | Analysis of PDEs 35Q30, 76N10 |
| url | https://arxiv.org/abs/2411.13896 |