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On the dimension of the singular set of solutions to the Navier–Stokes equations
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Robinson, James C. (James Cooper), 1969- and Sadowski, Witold. (2012) On the dimension of the singular set of solutions to the Navier–Stokes equations. Communications in Mathematical Physics, Vol.309 (No.2). pp. 497-506. ISSN 0010-3616
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Official URL: http://dx.doi.org/10.1007/s00220-011-1336-4
Abstract
In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of Lw(0T;Ls(R3)) , where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s − 1). We also show that if uLw(0T;Ls()) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s − 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, & Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QA Mathematics |
| Divisions: | Faculty of Science > Mathematics |
| Library of Congress Subject Headings (LCSH): | Navier-Stokes equations, Navier-Stokes equations -- Numerical solutions |
| Journal or Publication Title: | Communications in Mathematical Physics |
| Publisher: | Springer |
| ISSN: | 0010-3616 |
| Date: | January 2012 |
| Volume: | Vol.309 |
| Number: | No.2 |
| Page Range: | pp. 497-506 |
| Identification Number: | 10.1007/s00220-011-1336-4 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| URI: | http://wrap.warwick.ac.uk/id/eprint/39953 |
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