In this paper, we study the modified Navier–Stokes equations with a damping term, assuming that the initial data $u^0$ satisfies $\widehat{u^0}\in L^1({\mathbb{R}}^3)$. We establish the existence and uniqueness of a local solution, together with a blow-up criterion in the case where the maximal time of existence is finite. The analysis relies on standard techniques from Fourier analysis.