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Small scale formation for the 2-dimensional Boussinesq equation

Alexander Kiselev, Jaemin Park and Yao Yao

Vol. 18 (2025), No. 1, 171–198
Abstract

We study the 2-dimensional incompressible Boussinesq equations without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying [b]sup τ[0,t]ρ(τ)L2 tα for some α > 0. For the inviscid equation in the strip, we construct examples satisfying ω(t)L t3 and sup τ[0,t]ρ(τ)L t2 during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3-dimensional axisymmetric Euler equation whose velocity has infinite-in-time growth.

Keywords
small scale creation, 2-dimensional Boussinesq system
Mathematical Subject Classification
Primary: 35Q35, 76B03, 76D03
Milestones
Received: 1 December 2022
Revised: 6 June 2023
Accepted: 18 September 2023
Published: 15 December 2024
Authors
Alexander Kiselev
Department of Mathematics
Duke University
Durham, NC
United States
Jaemin Park
Departement Mathematik und Informatik
University of Basel
Basel
Switzerland
Department of Mathematics
Yonsei University
Seoul
South Korea
Yao Yao
Department of Mathematics
National University of Singapore
Singapore

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