The 2D Boussinesq equations with logarithmically supercritical velocities

2011/11/09 by Dongho Chae, Jiahong Wu, Chae, Dongho +1
Engineering · Mathematics · #35B35 #35B65 #35Q35 #76D03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1111.2082

openalex publication_date 2011/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq-Navier-Stokes equations. The velocity u in this system is related to the vorticity ω through the relations u=∇^⊥ ψ and Δψ= Λ^σ(log(I-Δ))^γω, which reduces to the standard velocity-vorticity relation when σ=γ=0. When either σ>0 or γ>0, the velocity u is more singular. The "quasi-velocity" v determined by ∇× v =ω satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when σ=0 and γ≥ 0. In addition, the vorticity ω is shown to be globally bounded in several functional settings such as L² for σ>0 in a suitable range.

Citations

• Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data
• On the global well-posedness for Boussinesq system
• Global regularity for the 2D Boussinesq equations with partial viscosity terms

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