A Lagrangian approach for the compressible Navier-Stokes equations

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in R^n, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) \emph{negative} indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from 0 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.