Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913923198877696 |
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| author | Jo, Min Jun |
| author_facet | Jo, Min Jun |
| contents | We study upper bounds on the box-counting dimension of the set of potential singular points in suitable weak solutions to the 3D incompressible hyperdissipative Navier-Stokes system
\begin{equation*} \partial_t u + (-Δ)^αu+(u\cdot \nabla)u+\nabla p = 0, \qquad \operatorname{div} u = 0, \end{equation*} for $α\in(1,5/4)$. Our main observation is that a classical iteration scheme developed in [11] and used in [27] to improve upper bounds for the full Laplacian case can be extended to the hyperdissipative case with properly chosen local quantities that are scale-invariant, despite non-locality of fractional Laplacian. This is achieved by matching up the correct orders of the temporal-spatial scales of the required estimates that effectively quantify $(-Δ)^α$ during the iterations. In particular, we adopt the hyperdissipative framework built in the recent breakthrough [5] where the upper bounds on the box-counting dimension of the set of potential singularities in $α$ are given by \begin{equation*}
L(α)= \frac{15-2α-8α^2}{3} \quad \mbox{for}\quad 1<α<\frac{5}{4}. \end{equation*} In this paper, we generalize the iteration scheme [27] designed for $α=1$ to the case $1<α<5/4$, which leads to the newly established bound \begin{equation*}
J(α)= \frac{36(3-α)(3+2α)(5-4α)}{-64α^3+272α^2-300α+369} \quad \mbox{for} \quad 1<α<\frac{5}{4}, \end{equation*} improving the aforementioned bound $L(α)$ obtained in [5]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_15274 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system Jo, Min Jun Analysis of PDEs 76D03, 76D05 We study upper bounds on the box-counting dimension of the set of potential singular points in suitable weak solutions to the 3D incompressible hyperdissipative Navier-Stokes system \begin{equation*} \partial_t u + (-Δ)^αu+(u\cdot \nabla)u+\nabla p = 0, \qquad \operatorname{div} u = 0, \end{equation*} for $α\in(1,5/4)$. Our main observation is that a classical iteration scheme developed in [11] and used in [27] to improve upper bounds for the full Laplacian case can be extended to the hyperdissipative case with properly chosen local quantities that are scale-invariant, despite non-locality of fractional Laplacian. This is achieved by matching up the correct orders of the temporal-spatial scales of the required estimates that effectively quantify $(-Δ)^α$ during the iterations. In particular, we adopt the hyperdissipative framework built in the recent breakthrough [5] where the upper bounds on the box-counting dimension of the set of potential singularities in $α$ are given by \begin{equation*} L(α)= \frac{15-2α-8α^2}{3} \quad \mbox{for}\quad 1<α<\frac{5}{4}. \end{equation*} In this paper, we generalize the iteration scheme [27] designed for $α=1$ to the case $1<α<5/4$, which leads to the newly established bound \begin{equation*} J(α)= \frac{36(3-α)(3+2α)(5-4α)}{-64α^3+272α^2-300α+369} \quad \mbox{for} \quad 1<α<\frac{5}{4}, \end{equation*} improving the aforementioned bound $L(α)$ obtained in [5]. |
| title | Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system |
| topic | Analysis of PDEs 76D03, 76D05 |
| url | https://arxiv.org/abs/2211.15274 |