An interpretation of Temam's extra force in the quasi-incompressible Navier-Stokes system

Abstract

We discuss the role of the extra force $$ \mathbf f_{\rm e}=-\frac 1 2(\nabla\cdot\mathbf v)\mathbf v $$ in the system of partial differential equations \begin{equation*} \left\{ \begin{aligned} &\frac{\partial\mathbf v}{\partial \rm t}+(\mathbf v\cdot\nabla)\mathbf v+\nabla \mathrm p-\frac 1 {\rm Re}\Delta\mathbf v=\mathbf f+\mathbf f_{\rm e},\\ &\frac 1 {\mathrm K}\frac{\partial \mathrm p}{\partial \rm t}+\nabla\cdot\mathbf v=0,\qquad \mathrm K>>1, \end{aligned}\right. \end{equation*} whose weak solution has been proved in [Arch. Rat. Mech. Analysis, 32:135-153] to approximate, in the limit $\mathrm K\to\infty$, the weak solution of the incompressible Navier-Stokes system. Taking the cue from [Ann. Mat. Pura Appl. 172:103-124], we provide a physical interpretation of the extra force $\mathbf f_{\rm e}$, showing that it is a manifestation of inertia.