Onsager critical solutions of the forced Navier-Stokes equations

TL;DR

This paper constructs new solutions to the 3D forced Navier-Stokes equations with vanishing viscosity that display anomalous dissipation and maintain uniform bounds in specific function spaces, addressing a key open question.

Contribution

It provides explicit examples of solutions with anomalous dissipation and uniform bounds, advancing understanding of the vanishing viscosity limit in fluid dynamics.

Findings

Solutions exhibit anomalous dissipation as viscosity vanishes

Solutions maintain uniform bounds in $L_t^3 C_x^{1/3 - ext{small}}$ spaces

Addresses an open question from previous research

Abstract

We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - \varepsilon}$, for any fixed $\varepsilon >0$. Our construction combines ideas of [BDL22] and [CCS22].