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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. 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. doi:10.1007/s00220-011-1336-4 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | January 2012 | ||||
Dates: |
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Volume: | Vol.309 | ||||
Number: | No.2 | ||||
Page Range: | pp. 497-506 | ||||
DOI: | 10.1007/s00220-011-1336-4 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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