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 ${\mathbb{R}}^n\mathrm{.}$ 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 $a_0:=\rho_0-1\in\dot B^{\frac np}{p,1},$ $u_0\in {\dot{B}}_{p,1}^{\frac{n}{p}-1}$ and $\theta_0\in\dot B^{\frac np-2}{p,1}.$ 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 $n\ge 2,$ and $1<p<2n\mathrm{.}$ 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).