Abstract: I will talk about various "mixing" questions that have attracted interest recently. For instance, "Can you stir your coffee to keep it hot for longer," or "How well can you stir cream into your coffee, and at what cost?"
Mathematically these questions translate into studying a negative Sobolev norm of a passively advected scalar. The study of such questions also involves very interesting connection Bressan's (still open!) rearrangement cost conjecture. I will spend most of the talk surveying recent results, and conclude with brief description of joint work with A. Kiselev, Xiaoqian Xu and myself.