# Some results on singular transport equations arising in fluid mechanics

# Some results on singular transport equations arising in fluid mechanics

We will discuss a few recent results in the study of fluid equations which stem from studying the dynamics of transport equations with non-local forcing. These are equations of the form: $f_t +u\cdot\nabla f =R(f)$ where $R$ is a singular integral operator and $u$ is a divergence-free vector field possibly depending upon $f$. These types of equations arise in a variety of physical scenarios. Depending upon the symbol of $R$, these equations can exhibit a number of different properties. We will briefly examine four different cases as they show up in different physical situations. The types of results we will discuss include: (1) Ill-posedness in critical spaces (2) Cascading solutions (3) Dispersion (4) "Inviscid damping" We will also mention several interesting open problems. Some of the results we will talk about are joint works with Nader Masmoudi and Klaus Widmayer.