On a non-solenoidal approximation to the incompressible Navier-Stokes equations

We establish an asymptotic profile that sharply describes the behavior as t -> infinity for solutions to a non-solenoidal approximation of the incompressible Navier-Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier-Stokes, for example, in L-loc(3)(R+ x R-3), provided epsilon -> 0, where epsilon > 0 is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier-Stokes for large times: indeed, its solutions can decay much slower as t -> infinity than the corresponding solutions of Navier-Stokes.