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Small scale formation
for the 2-dimensional Boussinesq equation
Alexander Kiselev, Jaemin Park and Yao Yao |
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Vol. 18 (2025), No. 1, 171–198
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Abstract |
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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 for some . For the inviscid equation in the strip, we construct examples satisfying and 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
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Mathematical Subject Classification
Primary: 35Q35, 76B03, 76D03
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Milestones
Received: 1 December 2022
Revised: 6 June 2023
Accepted: 18 September 2023
Published: 15 December 2024
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Authors |
| Alexander Kiselev | |
| Department of Mathematics Duke University Durham, NC United States |
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| 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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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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