Explicit singular steady profile for the dynamic rescaling 2D Boussinesq equations

Construct an explicit singular steady-state profile (\Omega, \Theta) with parameters (c_l, c_\omega) for the dynamic rescaling 2D Boussinesq system (equation \eqref{eqt:dynamic_rescaling_of_BS}) that can serve as a reference solution, analogous to the explicit singular profile available for the 1D Hou–Luo model.

Background

In the 1D Hou–Luo model, the authors construct an explicit weak singular steady state that guides and validates the numerical study of post-blowup dynamics in the dynamic rescaling variables.

For the 2D Boussinesq counterpart, while numerical evidence indicates convergence toward a singular profile with scaling ratio cl/cω≈−2c_l/c_\omega\approx -2, the authors report that they do not currently have an explicit analytic expression for such a singular steady profile to use as a reference in validating their numerical scheme.

References

Second, unlike the HL model, we are currently unable to obtain a steady singular profile of eqt:dynamic_rescaling_of_BS with explicit expressions that acts as a reference solution to validate our numerical scheme.

eqt:dynamic_rescaling_of_BS:

$\begin{aligned} &\Omega_{\tau} + (#1{U}+c_l#1{X})\cdot\nabla\Omega = c_{\omega}\Omega + \Theta_{X_1},\\ &\Theta_{\tau} + (#1{U}+c_l#1{X})\cdot\nabla\Theta = (c_l+2c_{\omega})\Theta,\\ &#1{U} = \nabla^{\perp}(-\Delta)^{-1}\Omega, \end{aligned} $

— Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation  (2604.01868 - Chen et al., 2 Apr 2026) in Subsection 6.1, Scenario 1 (Novel self-similar finite-time blowup of the 2D Boussinesq equations)