A Besov-based integration-by-parts method for the incompressible Navier-Stokes equations

2025/09/27 by Xinyu Cheng, Cheng, Xinyu, Zhaonan Luo +3
Computer Science · Engineering · #Numerical Methods and Algorithms #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.2509.23192

Abstract

This note introduces a novel numerical analysis framework for the incompressible Navier-Stokes equations based on Besov spaces. The key contribution of this note is to establish the stability and convergence of a semi-implicit time-stepping scheme by deriving precise error estimates in the B⁰_∞,1 and B⁰_∞,2 spaces. Another contribution of our analysis is the detailed treatment of the B⁰_∞,2 case, where a crucial integration-by-parts technique is employed to adeptly handle the nonlinear advection term. This technique allows for a refined estimate that effectively transfers derivatives onto the test functions, mitigating the inherent analytical challenges posed by the low regularity of these spaces. Our results provide sharper, more localized error bounds than in classical Sobolev spaces, directly linking the scheme's convergence to the critical regularity of the continuous solution. This work underscores the advantage of Besov spaces for the numerical analysis of nonlinear fluid PDEs.

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Related

• Non-convergence of the Navier-Stokes equations toward the Euler equations in the endpoint Besov spaces
• Space-time analyticity and refined analyticity radius of the Navier-Stokes equations in the critical Besov spaces
• Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces
• Robust error bounds for the Navier-Stokes equations using implicit-explicit second order BDF method with variable steps
• Asymptotic stability of the stationary Navier-Stokes flows in Besov spaces