Scholarlink

<p>In [Partially regular weak solutions of the Navier–Stokes equations in ℝ<sup>4</sup>×[0,∞), <i>Arch. Ration. Mech. Anal.</i>, 239(3):1771–1808, 2021], B. Wu discussed partial regularity results to the incompresible Navier–Stokes equations in ℝ<sup>4</sup> ×[0,+∞). To overcome the lack of compactness arising in the spatially 4-dimensional setting, the author introduced the notion of weak solution set involving concentration measures and proved the existence of partially regular weak solutions to the Navier–Stokes equations that satisfy some certain local energy inequalities. Moreover, Wu proved that the two-dimensional parabolic Hausdorff measure of the singular points of the weak solution set to the Navier–Stokes equations is finite.</p><p>In the current work, we are concerned with the Minkowski dimension of the set of potential singular points of this weak solution set to the four-dimensional Navier–Stokes equations. Taking full advantage of the partial regularity criteria given in the above-mentioned paper, we prove that the parabolic upper Minkowski dimension of the potential singular set is bounded by 31<i>/</i>11 (≈2<i>.</i>82). Moreover, we obtain an upper bound for the number of singular points at any fixed time.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

The Minkowski dimension and singular point numbers to the Navier–Stokes equations in ℝ4

  • 期刊论文

  • 发表日期：

    2025年5月26日

查看全文
  • Zhongbao Zuo

摘要

In [Partially regular weak solutions of the Navier–Stokes equations in ℝ4×[0,∞), Arch. Ration. Mech. Anal., 239(3):1771–1808, 2021], B. Wu discussed partial regularity results to the incompresible Navier–Stokes equations in ℝ4 ×[0,+∞). To overcome the lack of compactness arising in the spatially 4-dimensional setting, the author introduced the notion of weak solution set involving concentration measures and proved the existence of partially regular weak solutions to the Navier–Stokes equations that satisfy some certain local energy inequalities. Moreover, Wu proved that the two-dimensional parabolic Hausdorff measure of the singular points of the weak solution set to the Navier–Stokes equations is finite.

In the current work, we are concerned with the Minkowski dimension of the set of potential singular points of this weak solution set to the four-dimensional Navier–Stokes equations. Taking full advantage of the partial regularity criteria given in the above-mentioned paper, we prove that the parabolic upper Minkowski dimension of the potential singular set is bounded by 31/11 (≈2.82). Moreover, we obtain an upper bound for the number of singular points at any fixed time.