Partial Differential Equations / Mathematical Fluid Dynamics
Nonuniqueness of weak solutions to the Navier-Stokes equation
Buckmaster and Vicol resolve a decades-old open problem by proving that weak solutions to the 3D Navier-Stokes equations are not unique, using convex integration methods connected to turbulence theory.
Imagine you have a recipe (the Navier-Stokes equations) that's supposed to tell you exactly how a fluid like water will swirl and flow if you know how it starts. For a long time, mathematicians proved that this recipe always gives at least one valid answer, but they didn't know if it could give more than one different answer for the same starting point - kind of like asking a GPS for directions and getting two totally different valid routes to the same destination. This paper proves that, in certain mathematical settings, the equations actually CAN produce multiple different valid 'answers' (called weak solutions) starting from the same initial fluid state. The authors also show how these multiple solutions connect to real turbulent, chaotic fluid behavior - like the swirling patterns you see when you stir cream into coffee.
- 01
Proved nonuniqueness of finite kinetic energy weak solutions to the 3D Navier-Stokes equations, resolving a long-standing open question since Leray's 1934 existence result
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Constructed Hölder continuous dissipative weak solutions of the 3D Euler equations as a strong vanishing viscosity limit of finite energy Navier-Stokes weak solutions
- 03
Employed convex integration techniques adapted from the Euler equations to construct non-unique Navier-Stokes weak solutions
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Demonstrated a connection between turbulence phenomena, intermittency, and the inviscid limit in the context of weak solution nonuniqueness
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 Hölder 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.