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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.06657 |
| Etiquetas: |
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- In this article, we develop a new method, based on a time decomposition of a Cauchy problem elaborated in [6], to retrieve the well-known $L^\infty ([0,T],L^2(\mathbb{R}^d,\mathbb{R}^d))$ control of the solution of the incompressible Navier-Stokes equation in $\mathbb{R}^d$. We precisely explain how the Pythagorean theorem in $L^2(\mathbb{R}^d,\mathbb{R}^d)$ allows to get the proper energy estimate; however such an argument does not work anymore in $L^p(\mathbb{R}^d,\mathbb{R}^d)$, $p \neq 2$. We also deduce, by similar arguments, an already known $L^\infty ([0,T],L^1(\mathbb{R}^3,\mathbb{R}^3))$ control of vorticity for $d=3$.