2020
DOI: 10.48550/arxiv.2008.05802
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Partially regular weak solutions of the Navier-Stokes equations in $\mathbb{R}^4 \times [0,\infty[$

Bian Wu

Abstract: We show that for any given initial data and any external force, there exist partially regular weak solutions of the Navier-Stokes equations in R 4 which satisfy certain local energy inequalities and whose singular sets have locally finite 2-dimensional parabolic Hausdorff measure. With the help of a parabolic concentration-compactness theorem we are able to capture the lack of compactness arising in the spatially 4-dimensional setting by using defect measures, which we then incorporate into the partial regular… Show more

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“…In fact, Leray's structure theorem only holds up to d = 4 and the local energy inequality, a key ingredient in the partial regularity theory, remains absent in d ≥ 4 [DD07, Remark 1.1]. Despite such a difficulty, the existence of partially regular weak solutions in space-time was established in 4D [Sch77] by Scheffer and also a recent preprint [Wu20] by Wu. In dimension d ≥ 5 the existence of partially regular weak solutions( in space-time or in time) was unknown to our knowledge and Theorem 1.7 appears to be the first example of weak solutions with partial regularity in time in dimension d ≥ 5.…”
Section: Resultsmentioning
confidence: 99%
“…In fact, Leray's structure theorem only holds up to d = 4 and the local energy inequality, a key ingredient in the partial regularity theory, remains absent in d ≥ 4 [DD07, Remark 1.1]. Despite such a difficulty, the existence of partially regular weak solutions in space-time was established in 4D [Sch77] by Scheffer and also a recent preprint [Wu20] by Wu. In dimension d ≥ 5 the existence of partially regular weak solutions( in space-time or in time) was unknown to our knowledge and Theorem 1.7 appears to be the first example of weak solutions with partial regularity in time in dimension d ≥ 5.…”
Section: Resultsmentioning
confidence: 99%