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-\varepsilon\Delta u^\varepsilon+\nabla p^\varepsilon=0,\\ \mathrm{ div}\ 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.
Categories: math.AP
