Nonuniqueness of weak solutions to the Navier-Stokes equation

By T. Buckmaster, Vlad Vicol. Annals of Mathematics, 2017

443 citations recorded by the source corpus.

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4440332b72297ba0fb882bbbf78247c911f4ab97

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Convex integration and phenomenologies in turbulence (2019, methodology): “Third, we give a concise proof of the authors’ recent result [20], which proves the existence of infinitely many weak solutions of the NavierStokes in the regularity class C t L 2+ x ∩ C tW 1,1+ x .”
Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2021, methodology): “Buckmaster and Vicol [12] (see also [8]) demonstrated nonuniqueness of distributional solutions to the Cauchy problem for the Navier-Stokes equations via the convex integration method for any L(2) initial datum.”
Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces (2019, background): “[The recent result of Buckmaster & Vicol [4] shows that solutions in the sense of distributions need not satisfy the energy inequality, thereby proving also the nonuniqueness of such solutions.”
On solvability and ill-posedness of the compressible Euler system subject to stochastic forces (2017, background): “For completeness, let us mention that (1.4) may be seen as a “damping” term, the regularizing effect of which in the context of incompressible fluids has been recognized by Glatt–Holtz and Vicol [17], and for a general symmetric hyperbolic system by Kim [19].”
Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves (2020, methodology): “2]) and of the following nonuniqueness result at the PDE level, which in turn will be proved using “convex integration type” techniques borrowed from a groundbreaking work of Modena and Székelyhidi [24,25], improved later by Modena and Sattig [23](we refer to [9, 10, 15, 17, 18, 21] and the referenc”
On Ill‐ and Well‐Posedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations (2020, methodology): “The method was then further developed and successfully applied to a number of other fluid dynamics models, in particular to the Navier–Stokes system in three dimensions by Buckmaster and Vicol [14] and Buckmaster, Colombo, and Vicol [12], which permitted proving nonuniqueness for weak solutions not ”
Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 (2018, methodology): “4], and thus we omit most details where the estimates/proofs are mutatis-mutandis those from [5].”