Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation
This work addresses fundamental questions about singularity formation in fluid equations, which is crucial for understanding turbulence and stability in physics and applied mathematics, though it is incremental as it builds on prior numerical investigations.
The paper tackles the problem of identifying finite-time blowup scenarios in fluid dynamics models, specifically the 1D Hou-Luo model and 2D Boussinesq equations, by numerically demonstrating novel self-similar blowups with singular profiles and a two-stage feature involving local L∞ and Lp blowups at different times.
We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.