Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy

We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space ${\mathbb{R}}^2$ and on the upper half-plane ${\mathbb{R}}_{+}^2$, allowing infinite energy. In each case, we derive a one-dimensional reduction that captures the leading-order singular behavior of the original 2D system, and use a fixed-point argument to show the existence of finite-time self-similar blow-up solutions for the 1D systems. We also perform numerical simulations for verification and visualization.