Global well-posedness for the 3D compressible Navier-Stokes equations in optimal Besov space

2025/09/21 by Zihua Guo, Guo, Zihua, Zihao Song +3
Engineering · Mathematics · #35Q30 #76N06 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2509.17005

openalex publication_date 2025/09/21 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem to the 3D barotropic compressible Navier-Stokes equation. We prove global well-posedness, assuming that the initial data (ρ₀-1,u₀) has small norms in the critical Besov space \mathbbX_p=B_p,1^3/p(ℝ³)× B_p,1^-1+3/p(ℝ³) for 2≤ p<6 and (ρ₀-1,ρ₀u₀) satisfies an additional low frequency condition. Our results extend the previous results in \citeFD2010, CMZ2010, H20112 where p<4 is needed for high frequency, to the optimal range p<6. 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 semigroup that enables a L^q-framework for low frequency.

Citations

• On the well-posedness of the compressible Navier-Stokes equations
• Fourier analysis methods for the compressible Navier-Stokes equations
• A Lagrangian approach for the compressible Navier-Stokes equations

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