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

The present paper is the continuation of work [18], devoted to the study of an inviscid zero-Mach number system in the framework of endpoint   Besov spaces of type B∞,rs(Rd), r∈[1,∞]r∈[1,∞], d≥2d≥2, which can be embedded in the Lipschitz class C0,1C0,1. In particular, the largest case B∞,11 and the case of Hölder spaces C1,αC1,α are taken into account.The local in time well-posedness of this system is proved, under an additional finite-energy hypothesis on the initial data. The key to get this result is new a priori estimates for parabolic equations with variable coefficients in endpoint spaces B∞,rs(Rd), which are of independent interest.In the special case of space dimension d=2d=2, 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 refined a priori estimates in endpoint Besov spaces for transport equations.