Small scale creation in active scalars

An active scalar is advected by fluid velocity that is determined by the scalar itself. Active scalars appear in many situations in fluid mechanics, with the most classical example being 2D Euler equation in vorticity form. Usually, active scalar equations are both nonlinear and nonlocal, and their solutions spontaneously generate small scales. In this talk, I will discuss rigorous examples of small scale formation that involves infinite in time growth of derivatives for the 2D Euler equation, the SQG equation and the 2D IPM equation.