Publication

Lecture Notes on Intermittent Weak Solutions of the Three-dimensional Navier-Stokes Equations

2026

Abstract

This summary closely follows the paper Buckmaster et al. (Wild solutions of the Navier-Stokes equations whose singular sets in time have hausdorff dimension strictly less than 1, 2018. ArXiv:1809.00600) by the authors, which contains all the details regarding the specific gluing and convex integration technique used here. The lectures will be supplemented with material from the recent review paper Buckmaster and Vicol (EMS Surv. Math. Sci. 6(1/2):173–263, 2019), which contains the necessary background concerning weak solutions of the Euler and Navier-Stokes equations, their connection to phenomenological theories of fully developed turbulence, and the convex integration methods used to construct them.

Official source

https://infoscience.epfl.ch/entities/publication/3a63051d-8988-42de-9361-d7cb8f55ad87

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/3a63051d-8988-42de-9361-d7cb8f55ad87

About this result

Related concepts (17)

Navier–Stokes equations

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids.

Stokes flow

Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm.

Hausdorff dimension

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension.

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Lecture Notes on Intermittent Weak Solutions of the Three-dimensional Navier-Stokes Equations

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