Long-time well-posedness of analytic solutions to the Boussinesq boundary layer equations

Xiaolei Dong; Xiaolei Dong

doi:10.3934/era.2026361

Electronic Research Archive

2026, Volume 34, Issue 11: 8553-8568. doi: 10.3934/era.2026361

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Research article

Long-time well-posedness of analytic solutions to the Boussinesq boundary layer equations

Xiaolei Dong ^,

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

Received: 23 July 2026 Revised: 24 August 2026 Accepted: 29 September 2026 Published: 08 October 2026

This paper addresses the well-posedness of the two-dimensional Boussinesq boundary layer system without monotonicity constraints. Compared with the classical Prandtl equations, mutual velocity-temperature nonlinear coupling generates double derivative loss in both the momentum and thermal equations, which introduces new analytical obstacles beyond those appearing in the Prandtl setting and degenerates the solution regularity. To overcome this obstacle, we adopt analytic phase modulation and Gaussian-type decaying weights, construct anisotropic weighted Sobolev spaces, and combine refined nonlinear energy estimates and fractional damping absorption techniques. Under small initial data, we establish the long-time existence and uniqueness of analytic solutions, and derive the explicit solution lifespan scaling $ T_\varepsilon\sim\varepsilon^{-4/3} $.
- Boussinesq equations,
- boundary layer,
- well-posedness,
- analytic solutions
Citation: Xiaolei Dong. Long-time well-posedness of analytic solutions to the Boussinesq boundary layer equations[J]. Electronic Research Archive, 2026, 34(11): 8553-8568. doi: 10.3934/era.2026361

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Abstract

This paper addresses the well-posedness of the two-dimensional Boussinesq boundary layer system without monotonicity constraints. Compared with the classical Prandtl equations, mutual velocity-temperature nonlinear coupling generates double derivative loss in both the momentum and thermal equations, which introduces new analytical obstacles beyond those appearing in the Prandtl setting and degenerates the solution regularity. To overcome this obstacle, we adopt analytic phase modulation and Gaussian-type decaying weights, construct anisotropic weighted Sobolev spaces, and combine refined nonlinear energy estimates and fractional damping absorption techniques. Under small initial data, we establish the long-time existence and uniqueness of analytic solutions, and derive the explicit solution lifespan scaling $ T_\varepsilon\sim\varepsilon^{-4/3} $.

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