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 map from Xp to C([0,T]:Xp). In the previous works [Danchin, Commun. Partial Differ. Equ. 26, 1183–1233 (2001), Danchin, Nonlinear Differ. Equ. Appl. 12(1), 111–128 (2005), and Danchin, Ann. Inst. Fourier 64(2), 753–791 (2014)], the only known continuity of the solution map was from Xp−1 to C([0,T]:Xp−1) for 1 ⩽ p < d as a direct consequence of the uniqueness argument. To prove our results, we first extend the method of frequency envelope [see Tao, J. Hyperbolic Differ. Equ. 1(01), 27–49 (2004)] to the transport-parabolic setting. By this we obtained enhanced uniform estimates, in particular the uniform high-frequency smallness, for the sequence of smooth approximating solutions. Then we use the Lagrangian approach for the compressible Navier–Stokes equations [see Danchin, Ann. Inst. Fourier 64(2), 753–791 (2014)] to derive a new difference estimate for the velocity in Lt1Lx∞, which allows us to control the difference for low-frequency. As a by-product of our results, the Lagrangian transform (a,u)→(ā,ū)=(a◦X,u◦X) used in Danchin, Ann. Inst. Fourier 64(2), 753–791 (2014) is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.