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Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles
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Ożański, Wojciech (2020) Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles. Communications in Mathematical Physics, 374 . pp. 33-62. doi:10.1007/s00220-019-03588-0 ISSN 0010-3616.
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WRAP-weak-solutions-Navier-Stokes-inequality-arbitrary-energy-Ozanski-2019.pdf - Accepted Version - Requires a PDF viewer. Download (915Kb) | Preview |
Official URL: https://doi.org/10.1007/s00220-019-03588-0
Abstract
In a recent paper, Buckmaster and Vicol (Ann Math (2) 189(1):101–144, 2019) used the method of convex integration to construct weak solutions u to the 3D incompressible Navier–Stokes equations such that ∥u(t)∥L2=e(t) for a given non-negative and smooth energy profile e:[0,T]→R . However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray–Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier–Stokes inequality on R3 , that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier–Stokes equations). Given T>0 and a nonincreasing energy profile e:[0,T]→[0,∞) we construct weak solution to the Navier–Stokes inequality that are localised in space and whose energy profile ∥u(t)∥L2(R3) stays arbitrarily close to e(t) for all t∈[0,T] . Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier–Stokes equations, they satisfy the partial regularity theory of Caffarelli et al. (Commun Pure Appl Math 35(6):771–831, 1982). In fact, Scheffer’s constructions of weak solutions to the Navier–Stokes inequality with blow-ups (Commun Math Phys 101(1):47–85, 1985; Commun Math Phys 110(4): 525–551, 1987) show that the Caffarelli, Kohn & Nirenberg’s theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer’s. Namely, we obtain weak solutions to the Navier–Stokes inequality with both blow-up and a prescribed energy profile.
| 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 -- Numerical solutions, Arbitrary constants, Inequalities (Mathematics) | |||||||||
| Journal or Publication Title: | Communications in Mathematical Physics | |||||||||
| Publisher: | Springer | |||||||||
| ISSN: | 0010-3616 | |||||||||
| Official Date: | February 2020 | |||||||||
| Dates: |
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| Volume: | 374 | |||||||||
| Page Range: | pp. 33-62 | |||||||||
| DOI: | 10.1007/s00220-019-03588-0 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Communications in Mathematical Physics. The final authenticated version is available online at: https://doi.org/10.1007/s00220-019-03588-0 | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Copyright Holders: | © Springer-Verlag GmbH Germany, part of Springer Nature 2019 | |||||||||
| Date of first compliant deposit: | 8 May 2019 | |||||||||
| Date of first compliant Open Access: | 30 October 2020 | |||||||||
| RIOXX Funder/Project Grant: |
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