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.
Guo et al. (Sun,) studied this question.