Asymptotic stability of a singular steady state for the HL dynamic rescaling system

Establish the asymptotic stability of the singular steady state (\bar{\Omega}(X), \bar{\Theta}(X), \bar{c}_l, \bar{c}_\omega) = (\mathbf{1}_{\{X>1\}}/\sqrt{X-1}, \; \pi\,\mathbf{1}_{\{X>1\}}/2, \; 2, \; -1) of the dynamic rescaling Hou–Luo equations (equation \eqref{eqt:dynamic_rescaling_of_HL}), under suitable normalization conditions, by proving that for degenerate smooth initial data sufficiently close to the corresponding steady physical profiles in an appropriate norm, the rescaled solution (\Omega, \Theta) converges to (\bar{\Omega}, \bar{\Theta}) as the rescaled time \tau \to \infty.

Background

The authors construct a family of one-sided singular steady solutions for the 1D Hou–Luo model and its dynamic rescaling formulation, including an explicit profile (\bar{\Omega},\bar{\Theta},\bar{c}l,\bar{c}\omega)=(\mathbf{1}{{X>1}}/\sqrt{X-1},\;\pi\,\mathbf{1}{{X>1}}/2,\;2,\;-1). Numerical simulations suggest that, after a first L∞L^\infty blowup stage, solutions continue weakly and approach this singular steady profile in the rescaled variables.

Motivated by these observations, the paper posits a conjecture that this singular steady state is asymptotically stable under suitable normalization, meaning nearby degenerate smooth initial data evolve toward the same singular profile in the dynamic rescaling framework.

References

Based on these observations, we propose the following conjecture regarding the stability of $(\bar{\Omega},\bar{\Theta},\bar{c}l,\bar{c}{\omega})$. The singular steady state $(\bar{\Omega},\bar{\Theta},\bar{c}l,\bar{c}{\omega})$ is asymptotically stable. More specifically, under suitable normalization conditions, for any degenerate smooth initial data $(\omega0,\theta0)$ which is sufficiently close to $(\bar{\omega},\bar{\theta})$ in some norm $|\cdot|$, the solution of the Cauchy problem eqt:dynamic_rescaling_of_HL converges to $(\bar{\Omega},\bar{\Theta})$ in $|\cdot|$.

eqt:dynamic_rescaling_of_HL:

$\begin{aligned} &\Omega_\tau + (U+c_lX)\Omega_X=c_{}\Omega+\Theta_X,\\ &\Theta_\tau+ (U+c_lX)\Theta_X=(c_l+2c_{})\Theta,\\ &U_X =#1{H}(\Omega),\,\ U(0)=0, \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 “Scenario 1: novel asymptotically self-similar blowups with singular profiles.” (Section 2)