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 $mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $rho_0,$ velocity fields $u_0$ and temperatures $theta_0$ with $a_0:=rho_0-1indot B^{frac np}_{p,1},$ $u_0indot B^{frac np-1}_{p,1}$ and $theta_0indot B^{frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the emph{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 $ngeq2,$ 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 Danchin, Comm. Partial Differential Equations 26 (2001).