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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.00462 |
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| _version_ | 1866909214689984512 |
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| author | Melo, Wilberclay G. |
| author_facet | Melo, Wilberclay G. |
| contents | In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $α\in (0,\frac{5}{4})$ in $L^2(\mathbb{R}^3)$ and $\dot{H}^s(\mathbb{R}^3)$ ($s\leq0$). More precisely, we prove that $u$ satisfies the following upper bound: $$ \|u(t)\|_{2}^2\leq C(1+t)^{-\frac{3-2p}{2α}}, \quad\forall t>0.$$
This estimate leads us to show the next inequality: $$ \|u(t)\|_{\dot{H}^{-δ}}^2\leq C(1+t)^{-\frac{3-2δ-2p}{2α}}, \quad\forall t>0.$$ These results are obtained by applying standard Fourier Analysis and they hold for $α\in(0,\frac{5}{4}),$ $p\in[-1,\frac{3}{2})$, $δ\in [0, \frac{3-2p}{2})$ and $u_0\in L^2(\mathbb{R}^3)\cap \mathcal{Y}^p(\mathbb{R}^3)$ (and also $u_0\in L^1(\mathbb{R}^3)$ for $p=-1$ and a certain finite set of values of $α$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00462 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Temporal decay rates for weak solutions of the Navier-Stokes Equations with supercritical fractional dissipation Melo, Wilberclay G. Analysis of PDEs 35Q30, 35Q35, 76D05, 35B40 In this paper, we establish temporal decay for a weak solution $u(x,t)$ (with initial data $u_0$) of the Navier-Stokes equations with supercritical fractional dissipation $α\in (0,\frac{5}{4})$ in $L^2(\mathbb{R}^3)$ and $\dot{H}^s(\mathbb{R}^3)$ ($s\leq0$). More precisely, we prove that $u$ satisfies the following upper bound: $$ \|u(t)\|_{2}^2\leq C(1+t)^{-\frac{3-2p}{2α}}, \quad\forall t>0.$$ This estimate leads us to show the next inequality: $$ \|u(t)\|_{\dot{H}^{-δ}}^2\leq C(1+t)^{-\frac{3-2δ-2p}{2α}}, \quad\forall t>0.$$ These results are obtained by applying standard Fourier Analysis and they hold for $α\in(0,\frac{5}{4}),$ $p\in[-1,\frac{3}{2})$, $δ\in [0, \frac{3-2p}{2})$ and $u_0\in L^2(\mathbb{R}^3)\cap \mathcal{Y}^p(\mathbb{R}^3)$ (and also $u_0\in L^1(\mathbb{R}^3)$ for $p=-1$ and a certain finite set of values of $α$). |
| title | Temporal decay rates for weak solutions of the Navier-Stokes Equations with supercritical fractional dissipation |
| topic | Analysis of PDEs 35Q30, 35Q35, 76D05, 35B40 |
| url | https://arxiv.org/abs/2406.00462 |