On the well-posedness of the full compressible Navier-Stokes system in critical Besov spaces

We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in Rn. We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities Θ₀, velocity fields u₀ and temperatures θ₀ with a0:=Θ0-1∈B˙p,1np, u0∈B˙p,1np-1 and θ0∈B˙p,1np-2. After recasting the whole system in Lagrangian coordinates, and working with the total energy along the flow rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is n≥2, and 1<p<2n. Back to Eulerian coordinates, this allows to improve the range of p's for which the system is locally well-posed, compared to [7].