Abstract: 

Many active scalar fluid equations are locally-in-time well-posed in Hölder spaces C^k,α (k ≥ 1, 0 < α < 1), by the classical particle trajectory method. On the contrary, in the endpoint C^k spaces (k ≥ 1), there are examples of initial data in C^k for which no solution remains bounded in C^k, even for a short time. In this talk, we will unpack the mechanisms behind ill-posedness with the example of the Incompressible Porous Media equation, which exhibits this phenomenon despite its partial dissipation effects.