Single exponential $H^1$-upper bounds for the primitive equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918288507797504 |
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| author | Kashiwabara, Takahito |
| author_facet | Kashiwabara, Takahito |
| contents | The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $Ω$, which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution $v$ with initial data $a$, we establish \emph{a priori} bounds in $L^\infty(0, \infty; H^1(Ω)) \cap L^2(0, \infty; \dot H^2(Ω))$, the exponential part of which is $\exp(C \|a\|_{L^2(Ω)}^2)$. This is in contrast to the upper bounds reported in the existing literature that are double exponential. Furthermore, the uniform-in-time estimate for the Neumann condition case, in which the Poincaré inequality is unavailable for $v$, seems to be new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09183 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Single exponential $H^1$-upper bounds for the primitive equations Kashiwabara, Takahito Analysis of PDEs The three dimensional primitive equations with full viscosity are considered in a horizontally periodic box $Ω$, which are subject to either the homogeneous Neumann or Dirichlet conditions on the upper and bottom parts of the boundary. For a strong solution $v$ with initial data $a$, we establish \emph{a priori} bounds in $L^\infty(0, \infty; H^1(Ω)) \cap L^2(0, \infty; \dot H^2(Ω))$, the exponential part of which is $\exp(C \|a\|_{L^2(Ω)}^2)$. This is in contrast to the upper bounds reported in the existing literature that are double exponential. Furthermore, the uniform-in-time estimate for the Neumann condition case, in which the Poincaré inequality is unavailable for $v$, seems to be new. |
| title | Single exponential $H^1$-upper bounds for the primitive equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.09183 |