On regularity of the Euler equations in fluid dynamics

Abstract

We assert that the solutions to the Cauchy problem of the inviscid vorticity equation remain regular and unique for any smooth initial data of finite energy. However, the primitive formulation of the Euler equations is not well-posed, due to the passive pressure. One of the implications is that the anomalous energy dissipation, anticipated by Onsager (1949), cannot occur in inviscid flows. In the complete absence of viscous effects, the ultimate accumulation of enstrophy in sealed domains is bound to become arbitrarily excessive, if there is a sustained supply of shears and strains.