Summary: We are interested in the existence and global uniqueness of the solution to the flow equation of an inhomogeneous fluid, when the initial velocity is in the critical homogeneous Besov space \(B_{p,1}^{-1+\frac{N}{p}}(\mathbb R^{ N })\). Let us note that this result followed from the results of H. Abidi which generalized the work of R. Danchin. However, the existence of solutions was obtained when \(1<p<2N\) and uniqueness was shown under the more restrictive assumption \(1<p\leqslant N\). Our result resolves the question of existence for all \(1<p<+\infty \) and uniqueness for \(1<p\leqslant 2N\). As an interesting application of this theorem, we obtain global existence for oscillating initial data.