Analysis of an inviscid zero-Mach number system in endpoint Besov spaces with finite-energy initial data

The present paper is the continuation of work [14], devoted to the study of an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov spaces of type $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$, which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the largest case $B^1_{\infty,1}$ and the case of H\"older spaces $C^{1,\alpha}$ are permitted. The local in time well-posedness result is proved, under an additional $L^2$ hypothesis on the initial inhomogeneity and velocity field. A new a priori estimate for parabolic equations in endpoint spaces $B^s_{\infty,r}$ is presented, which is the key to the proof. In dimension two, we are able to give a lower bound for the lifespan, such that the solutions tend to be globally defined when the initial inhomogeneity is small. There we will show a refined a priori estimate in endpoint Besov spaces for transport equations with \emph{non solenoidal} transport velocity field.