| Abstract: |
| We consider the 2D Boussinesq system on sectors with angle less than $\pi$, and show that there exists Lipschitz continuous velocity field and density pair $(u_0, \rho_0)$ which becomes singular in finite time. The initial data can be compactly supported and in particular the solution has finite energy. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of exactly scale-invariant solutions, and finally a cut-off argument. We also discuss implications of these results to the issue of singularity formation for the 3D Euler equations in the axi-symmetric geometry. |
|