A global regularity result for the 2D Boussinesq equations with critical dissipation

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 137; no. 1; pp. 269 - 290
• Main Authors: Stefanov, Atanas, Wu, Jiahong
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.03.2019
• Subjects: Abstract Harmonic Analysis; Analysis; Dynamical Systems and Ergodic Theory; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-018-0073-4

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by Λ α u in the velocity equation and by Λ β θ in the temperature equation, where Λ − − Δ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when ( α , β ) is in the critical range: α > ( 1777 − 23 ) / 24 = 0.789103... , β > 0, and α + β = 1. This result improves previous work which obtained the global regularity for α > ( 23 − 145 ) / 12 ≈ 0.9132 , β > 0 , and α + β = 1.