Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?

By Hao Jia, Vladim'ir vSver'ak. 2013

151 citations recorded by the source corpus.

An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in \cite{JiaSverak}. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditions seems to be approachable by relatively straightforward numerical simulations which involve only smooth functions.

Subfields: Applied Mathematics, Computational Mechanics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/22a584b72112b69b7771cf44c7caac8e7ddbb7b6

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In Applied Mathematics

In Computational Mechanics

In Control and Systems Engineering

Citations in the corpus

Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, background): “A very interesting conjecture of Jia-Šverák [22, 23] essentially states that the Prodi-Serrin uniqueness criteria are sharp, and that the non-uniqueness of Leray-Hopf weak solutions may already be expected in the regularity class L∞t L 3,∞ x .”
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2021, background): “In [33], spectral conditions on the linearized operator L (σ) ss (that is, linearized around the profiles Uσ) are identified for such bifurcations to occur.”
Convex integration and phenomenologies in turbulence (2019, methodology): “8 via a bifurcation at t = 0 – this is indeed the strategy employed by Šverák and Jia [105].”
Nonuniqueness of weak solutions to the Navier-Stokes equation (2017, unknown): “Thus, the works [29, 23] strongly suggest that the Ladyzhenskaya-Prodi-Serrin regularity criteria are sharp.”