Closed-form finite-time blow-up and stability for a $(1+2)$D system (E1) derived from the 2D inviscid Boussinesq equations
Yaoming Shi
Abstract
In polar variables $(x,θ)$ on a planar sector, we study a $(1+2)$D system (E1) derived from the two-dimensional inviscid Boussinesq equations. Under a parity/symmetry ansatz on the whole plane (odd/even reflection across the axes), we show that the velocity-pressure form of the 2D inviscid Boussinesq system admits an exact reformulation in terms of Hou--Li type new variables $(u,v,g)$. In the reformulated system (E1), the vortex stretching terms are greatly simplified $(uv,v^2-u^2,-g^2)$. This prompts us to treat $(u,v,g)$ as the \textbf{vorticity building blocks}. Our first main result is the discovery of explicit \emph{smooth} solutions that blow up in finite time $0<T<\infty$ while a natural weighted energy remains \emph{uniformly bounded} for all $t\in[0,T]$. The construction proceeds in three steps. (1) We identify special \emph{ridge rays} $θ_0=\pmπ/4$ such that, under the divergence-free constraint, system (E1) reduces on each ridge to a $(1+1)$D Constantin--Lax--Majda type \emph{convection-free} reaction system in $(t,x)$; (2) We then embed these $(1+1)$D closed-form ridge solutions into the full sector $x\in[0,\infty)$, $θ\in[-π/4,π/4]$ by introducing carefully tuned $θ$-dependent seed data, producing an explicit background profile that blows up only at $(x,θ)=(0,\pmπ/4)$. (3) Finally, we derive the perturbation equations around this background and prove \emph{linear and nonlinear stability} in high-regularity weighted Sobolev norms. Consequently, the constructed background profiles are \emph{stable finite-time blow-up solutions} of (E1).