Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918152268414976 |
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| author | Galdi, Giovanni Paolo |
| author_facet | Galdi, Giovanni Paolo |
| contents | Let $\bfu$ be a Leray solution to the Navier-Stokes boundary-value problem in an exterior domain, vanishing at infinity and satisfying the generalized energy inequality. We show that if there exist $R>0$ and ${\sf s}\ge \frac23 q$, $q>6$, such that the $L^{\sf s}-$norm of $\bfu$ on the spherical surface of radius $R$ divided by $R$ is less than a constant depending only on {\sf s} and $q$, then $\bfu(x)$ must decay as $|x|^{-1}$ for $|x|\to\infty$. This result is proved with an approach based on a new theory of very weak solutions in exterior domains which, as such, is of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations Galdi, Giovanni Paolo Analysis of PDEs Mathematical Physics Let $\bfu$ be a Leray solution to the Navier-Stokes boundary-value problem in an exterior domain, vanishing at infinity and satisfying the generalized energy inequality. We show that if there exist $R>0$ and ${\sf s}\ge \frac23 q$, $q>6$, such that the $L^{\sf s}-$norm of $\bfu$ on the spherical surface of radius $R$ divided by $R$ is less than a constant depending only on {\sf s} and $q$, then $\bfu(x)$ must decay as $|x|^{-1}$ for $|x|\to\infty$. This result is proved with an approach based on a new theory of very weak solutions in exterior domains which, as such, is of independent interest. |
| title | Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2510.00265 |