On the energy dissipation phenomena for Navier–Stokes equations of compressible fluid flow

This paper is concerned with the Cauchy problem of compressible Navier–Stokes equations. Both the anomalous energy dissipation and the vanishing global dissipation are surveyed. First, we construct a family of smooth solutions which exhibit anomalous dissipation when the viscous coefficient $\epsilon$ tends to zero. Second, assume that the weak solutions have additional (uniformly in $\epsilon$ ) regularity, then the convergence rate of vanishing global dissipation is proportional to a power function of $\epsilon$ . The results indicate that the inviscid singularity is caused by the lack of smoothness of solutions, not the viscosity.