A survey on some vanishing viscosity limit results

We present a survey concerning the convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations. After considering the Cauchy problem, particular attention is given to the convergence under Navier slip-type boundary conditions. We show that, in the presence of flat boundaries (typically, the half-space case), convergence holds, uniformly in time, with respect to the initial data's norm. In spite of this result (and of a similar result for arbitrary two-dimensional domains), strong inviscid limit results are proved to be false in general domains, in correspondence to a very large family of smooth initial data. In Section 6, we present a result in this direction.

Automated summary

This paper investigates the convergence of solutions to the three-dimensional evolutionary Navier-Stokes equations to solutions of the Euler equations as the viscosity tends to zero. It focuses on the convergence under Navier slip-type boundary conditions after considering the Cauchy problem. It is shown that, in the presence of flat boundaries (such as the half-space case), convergence holds uniformly in time with respect to the initial data's norm. However, strong inviscid limit results are proven to be false in general domains corresponding to a large family of smooth initial data. A result in this direction is presented in Section 6.