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

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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_ε ∼ ε^−4/3.

Keywords

Boussinesq equations / boundary layer / well-posedness / analytic solutions

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