Anomalous Dissipation for the d-dimensional Navier-Stokes Equations

The purpose of this paper is to study the vanishing viscosity limit for the d-dimensional Navier–Stokes equations in the whole space: \begin{equation*} \begin{cases} \partial_tu^\varepsilon+u^\varepsilon\cdot \nabla u^\varepsilon-\varepsilonΔu^\varepsilon+\nabla p^\varepsilon=0,\ $\mathrm{d}\mathrm{i}\mathrm{v}$\ u^\varepsilon=0. \end{cases} \end{equation*} We aim to presenting a simple rigorous examples of initial data which generates the corresponding solutions of the Navier–Stokes equations do exhibit anomalous dissipation. Precisely speaking, we show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero.