[논문 리뷰] The low mach number limit of global solutions to the full compressible Navier-Stokes system in critical Besov spaces with large initial data

Sai Li|arXiv (Cornell University)|2026. 03. 01.

Navier-Stokes equation solutions인용 수 0

한 줄 요약

본 논문은 대규모 초기 데이터에 대해 임계 Besov 공간에서 전체 압축성 Navier–Stokes 방정식의 규칙적인 해의 전역 존재성을 증명하고, 저 Mach 수 한계가 비압축 Boussinesq 시스템으로 수학적으로 엄밀하게 정당화됨을 보인다.

ABSTRACT

We are concerned with global existence of regular solutions to full compressible Navier-Stokes equations and their asymptotic behavior when the Mach number is sufficiently small. We establish global existence in critical Besov spaces for arbitrary large initial date provided that the divergence-free component of initial velocity and the difference between initial temperature and density generate a global regular solution to incompressible Boussinesq systems. Moreover, we rigorously justify the convergence to the incompressible model as the Mach number tends to zero. The proof relies on a fine-grained analysis of the high-middle-low frequencies of density, velocity and temperature. Our result can be seen as an improvement on Danchin and He [Math. Ann., 366 (2016), no. 3-4, pp. 1365-1402], including the extension from small initial data to large initial data and new convergence results which hold at the level of critical regularity.

연구 동기 및 목표

• Establish global well-posedness in critical Besov spaces for the full compressible Navier–Stokes system with large initial data.
• Characterize the asymptotic behavior as the Mach number tends to zero and prove convergence to incompressible models.
• Link the low Mach limit to the Boussinesq system via precise frequency-based analysis.

제안 방법

• Employ a three-part high/middle/low frequency analysis in critical Besov spaces.
• Define and use energy functional E^{ε} and the quantity M^{ε,α}_{p,q}[a,u,θ;τ,σ](I) to control solutions.
• Introduce the variables τ^{ε} and σ^{ε} to handle singular terms and decompose velocity into divergence-free and curl-free components using projectors Q and P.
• Split the dynamics into the incompressible limit (Θ, v) solving the Boussinesq system and the dispersive decay of wave-like components (τ^{ε}, Q u^{ε}).
• Prove uniform-in-ε a priori estimates and obtain convergence results at critical regularity.

실험 결과

연구 질문

• RQ1 Can global regular solutions to the full compressible Navier–Stokes equations be obtained in critical Besov spaces for arbitrarily large initial data under Mach number scaling?
• RQ2 Does the solution converge to the incompressible (Boussinesq) model as the Mach number ε tends to zero, and at what regularity level?
• RQ3 What is the precise mechanism (in frequency space) that enables the low Mach limit to hold for large data?
• RQ4 How do the divergence-free and curl-free components of the velocity field behave in the low Mach limit?
• RQ5 What are the necessary uniform estimates that ensure the convergence results hold at the level of critical regularity?

주요 결과

• Global existence of a unique solution (a^{ε}, u^{ε}, θ^{ε}) in the stated Besov-class framework for 0 < ε ≤ ε_{0}, with large initial data.
• Convergence to the incompressible Boussinesq system as ε → 0, with explicit convergence statements for the divergence-free part of the velocity.
• Quantitative bounds showing ε-scaled norms of density and temperature fluctuations remain controlled, enabling uniform-in-ε estimates.
• Establishment of convergence at the level of critical regularity, improving prior results that were restricted to small data.
• A three-part frequency analysis strategy that handles high, middle, and low frequencies to manage nonlinear interactions and dispersive effects.