Francesca Crispo

A missed persistence property for the Euler equations and its effect on inviscid limits

We consider the problem of strong convergence, as the viscosity goes to zero, of the solutions to the three-dimensional evolutionary Navier–Stokes equations, under the Navier slip-type boundary condition (1.4), to the solution of the Euler equations under the no-penetration condition. In two dimensions, the above strong convergence holds in any smooth domain. Furthermore, in three dimensions, arbitrarily strong convergence results hold in the half-space case. In spite of the above results, recently we presented an explicit family of smooth initial data in the 3D sphere, for which the result fails. The result was proved as a by-product of the lack of time persistency for the above boundary condition under the Euler flow. Our aim here is to show a more general, and simpler proof, displayed in arbitrary, smooth domains.

On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem

We prove space–time decay estimates of suitable weak solutions to the Navier–Stokes Cauchy problem, corresponding to a given asymptotic behavior of the initial data of the same order of decay. We use two main tools. The first is a result obtained in [7] for the behavior of the solution in a neighborhood of t = 0 in the -norm, which enables us to furnish a representation formula for a suitable weak solution. The second is the asymptotic behavior of for . Following Lerays point of view, roughly speaking our result proves that a possible space–time turbulence does not perturb the asymptotic spatial behavior of the initial data of a suitable weak solution.