  Even though the Madelung equations are central to many 'classical' approaches
to the foundations of quantum mechanics such as Bohmian and stochastic
mechanics, no coherent mathematical theory has been developed so far for this
system of partial differential equations. Wallstrom prominently raised
objections against the Madelung equations, aiming to show that no such theory
exists in which the system is well-posed and in which the Schr\"odinger
equation is recovered without the imposition of an additional 'ad hoc
quantization condition'--like the one proposed by Takabayasi. The primary
objective of our work is to clarify in which sense Wallstrom's objections are
justified and in which sense they are not, with a view on the existing
literature. We find that it may be possible to construct a mathematical theory
of the Madelung equations which is satisfactory in the aforementioned sense,
though more mathematical research is required.
