  Many equations that model fluid behaviour are derived from systems that
encompass multiple physical forces. When the equations are written in non
dimensional form appropriate to the physics of the situation, the resulting
partial differential equations often contain several small parameters. We study
a general class of such PDEs called active scalar equations which in specific
parameter regimes produce certain well known models for fluid motion. We
address various mathematical questions relating to well-posedness, regularity
and long time behaviour of the solutions to this general class including
vanishing limits of several diffusive parameters.

Analysis of PDEs