Title: Well-posedness for the compressible Navier-Stokes equations in critical Besov space

Abstract: The Cauchy problem to the barotropic compressible Navier-Stokes equation in critical Besov spaces is considered. I will talk about two recent results obtained in (I) arXiv:2409.01031 and (II) arXiv:2509.17005.

I. Local well-posedness

We prove the continuity of the solution map by combining the frequency envelope method and the Lagrangian approach. This result bridges the Eulerian and Lagrangian methods in the study of compressible Navier-Stokes equation.

II. Global well-posedness

We prove global well-posedness with small data in the optimal critical Besov space assuming some low frequency condition on the initial density and mo mentum. The main ingredients of the proof consist of: a novel nonlinear transform that uses momentum formulation for low-frequency and effective velocity method for high frequency, and estimate of parabolic-dispersive semi group that enables a Lq-framework for low frequency.