Title: Blow-up, gradient growth, and regularization in 1D active scalars

Abstract: There are many simplified 1D equations used to model 3D Euler and 2D surface quasi-geostrophic equations. We will review recent results on these equations. Using the trajectory map and estimates on the nonlocal quantities, we will prove double exponential growth for the
1D Euler model and blow-up for the 1D alpha-patch model. Lastly,
we will discuss possible regularization of the Cordoba-Cordoba-
Fontelos model in the supercritical regime.