Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910707206848512 |
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| author | Honoré, Igor |
| author_facet | Honoré, Igor |
| contents | We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that $\sup_{t \in [0,T]} \int_{0}^t (t-s)^{-θ} \|\nabla^k u(s,\cdot)\|_{L^r} ds<+ \infty$, for all $θ< \frac{3-kr}{2r}$, $k \in [0,2]$, and $1<r<\frac{3}{k}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces Honoré, Igor Analysis of PDEs We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that $\sup_{t \in [0,T]} \int_{0}^t (t-s)^{-θ} \|\nabla^k u(s,\cdot)\|_{L^r} ds<+ \infty$, for all $θ< \frac{3-kr}{2r}$, $k \in [0,2]$, and $1<r<\frac{3}{k}$. |
| title | Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.13980 |