Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912244146634752 |
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| author | Hu, Ruilin Nguyen, Phuoc-Tai Nguyen, Quoc-Hung Zhang, Ping |
| author_facet | Hu, Ruilin Nguyen, Phuoc-Tai Nguyen, Quoc-Hung Zhang, Ping |
| contents | In this paper, we study the quantitative regularity and blowup criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space framework. Specifically, we consider solutions $u\in L^\infty_t(\dot{B}_{p,\infty}^{-1+\frac{3}{p}})$ such that $|D|^{-1+\frac{3}{p}}|u|\in L^\infty_t (L^p)$ with $3<p<\infty$. By deriving refined regularity estimates and substantially improving the strategy in \cite{Tao_20}, we overcome difficulties stemming from the low regularity of the Besov spaces and establish quantitative bounds for such solutions. These bounds are expressed in terms of a triple exponential of $\| u (t)\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ combined with a single exponential of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$. Consequently, we obtain a new blowup rate which can be interpreted as a coupling of triple logarithm of $\| u(t) \|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ and a single logarithm of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06483 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces Hu, Ruilin Nguyen, Phuoc-Tai Nguyen, Quoc-Hung Zhang, Ping Analysis of PDEs 35Q30, 37N10, 76D05 In this paper, we study the quantitative regularity and blowup criteria for classical solutions to the three-dimensional incompressible Navier-Stokes equations in a critical Besov space framework. Specifically, we consider solutions $u\in L^\infty_t(\dot{B}_{p,\infty}^{-1+\frac{3}{p}})$ such that $|D|^{-1+\frac{3}{p}}|u|\in L^\infty_t (L^p)$ with $3<p<\infty$. By deriving refined regularity estimates and substantially improving the strategy in \cite{Tao_20}, we overcome difficulties stemming from the low regularity of the Besov spaces and establish quantitative bounds for such solutions. These bounds are expressed in terms of a triple exponential of $\| u (t)\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ combined with a single exponential of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$. Consequently, we obtain a new blowup rate which can be interpreted as a coupling of triple logarithm of $\| u(t) \|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}$ and a single logarithm of $\bigl\| |D|^{-1+\frac{3}{p}}|u(t)| \bigr\|_{L^p}$. |
| title | Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces |
| topic | Analysis of PDEs 35Q30, 37N10, 76D05 |
| url | https://arxiv.org/abs/2411.06483 |