On the well-posedness of the compressible Navier-Stokes equations

We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space Xₚ=Ḃp,1d/p× Ḃp,1-1+d/p for 1≤ p<2d with d≥2. The main new result is the continuity of the solution maps from Xₚ to C([0,T]: Xₚ), which was not proved in previous works {D2001, D2005, D2014}. To prove our results, we combine the method of frequency envelope (see {Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see {D2014}). As a by-product, the Lagrangian transform (a,u)→ ( a, u)=(a∘ X, u∘ X) used in {D2014} is a continuous bijection and hence bridges the two methods using Eulerian and Lagrangian coordinates.