Publication

Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions

2023

Abstract

We exhibit non-unique Leray solutions of the forced Navier–Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in Albritton et al. (Ann Math 196(1):415–455, 2022), the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as t→ 0 + . In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik (Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018; Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018) to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in Albritton et al. (2022).

Official source

https://infoscience.epfl.ch/entities/publication/5ef39e4f-5ae5-4c61-8fc7-d46f2af48ddf

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5ef39e4f-5ae5-4c61-8fc7-d46f2af48ddf

About this result

Related concepts (23)

Euler equations (fluid dynamics)

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity. The Euler equations can be applied to incompressible or compressible flow. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field.

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.

Related publications (1)

Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions

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