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Weak Solutions to the Navier–Stokes Inequality with Arbitrary Energy Profiles

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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 \(\Vert u(t) \Vert _{L^2} =e(t)\) for a given non-negative and smooth energy profile \(e:[0,T]\rightarrow \mathbb {R}\). 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 \(\mathbb {R}^3\), 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 \(e:[0,T] \rightarrow [0,\infty )\) we construct weak solution to the Navier–Stokes inequality that are localised in space and whose energy profile \(\Vert u(t)\Vert _{L^2 (\mathbb {R}^3 )}\) stays arbitrarily close to e(t) for all \(t\in [0,T]\). 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.

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Notes

  1. Note that here we use the convention of “nonanticipating” cylinders; namely that Q is based at a point (x, t) when (x, t) lies on the upper lid of the cylinder.

  2. In fact, (1.7) implies a stronger estimate than \(d_H (S) \le 1\); namely that \({\mathcal {P}}^1(S)=0\), where \({\mathcal {P}}^1(S)\) is the parabolic Hausdorff measure of S (see Theorem 16.2 in Robinson et al. [16] for details).

  3. Note that the point \(x\in U\) at which the right-hand side of (4.2) will become negative is located close to the \(\partial U\) since only for such x\(\phi (x) =1\) but \(f(x) <\max \,f\).

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Acknowledgements

The author is very grateful to James Robinson for his careful reading of a draft of this article and his numerous comments, which significantly improved its quality. The author was supported partially by EPSRC as part of the MASDOC DTC at the University of Warwick, Grant No. EP/HO23364/1, and partially by postdoctoral funding from ERC 616797.

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  1. Department of Mathematics, University of Southern California, Los Angeles, USA

    Wojciech S. Ożański

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Communicated by C. De Lellis

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Ożański, W.S. Weak Solutions to the Navier–Stokes Inequality with Arbitrary Energy Profiles. Commun. Math. Phys. 374, 33–62 (2020). https://doi.org/10.1007/s00220-019-03588-0

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