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

Lorenzo Brandolese (1)

1 EDPA - Équations aux dérivées partielles, analyse

We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ 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, e.g., in $L^3_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, 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 $t\to+\infty$ than the corresponding solutions of Navier–Stokes.

Mots clés: long-time asymptotics; Asymptotic profiles; artificial compressibility

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brandolese. On a non-solenoidal approximation to the incompressible Navier-Stokes equations. J. London Math. Soc. (2), 2017, 96 (2), pp.326-344. ⟨hal-01522952⟩