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

2024/09/02 by Guo, Zihua, Yang, Minghua, Zhang, Zeng · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.01031

Abstract

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 \mathbbX_p=B_p,1^(d)/(p)× B_p,1^-1+(d)/(p) for 1≤ p<2d with d≥2. The main new result is the continuity of the solution maps from \mathbbX_p to C([0,T]: \mathbbX_p), which was not proved in previous works \citeD2001, D2005, D2014. To prove our results, we derive a new difference estimate in L_t¹L_x^∞. Then we combine the method of frequency envelope (see \citeTao04) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \citeD2014). As a by-product, the Lagrangian transform (a,u)→ ( a, u)=(a∘ X, u∘ X) used in \citeD2014 is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.

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