A scaling and energy equality preserving approximation for the 3D Navier–Stokes equations in the finite energy case

We discuss a new model (inspired by the work of Vishik and Fursikov) approximating the 3D Navier–Stokes equations, which preserves the scaling as in the Navier–Stokes equations and thus allows the study of self-similar solutions. Using some energy estimates and Leray’s limiting process, we show the existence of a solution of this model in the finite energy case, and the energy equality and inequality fulfilled by it. This approximation can be shown to converge to the Navier–Stokes equations using a mild approach based on the approximated pressure, and the solution satisfies Scheffer’s local energy inequality, an essential tool for proving Caffarelli, Kohn and Nirenberg’s regularity criterion. We also give a partial result of self-similarity satisfied by the approximated solution in the infinite energy case.