Question

In physics and engineering sources, calculus-based formalisms - whether differential forms on a manifold, or "differentials" of functions of several variables - are presented as a way of modeling and reasoning about thermodynamic systems. However, I've found little to no mathematical background given to justify these formal manipulations when there are phase changes or other discontinuities. In such regions, one would expect the theory to break down since partial derivatives and differential forms are not defined.

Nevertheless, if you "shut up and calculate", everything seems to work out fine. Why is this the case, and what is the proper mathematical state space and framework for systems with phase transitions or other non-smooth properties? (perhaps some sort of "weakly-differentiable" manifold?)

Answer 1

Most physical systems modeled by PDEs can be transformed to an integral weak form where the smoothness requirements of the unknowns are lower (*).

For example, although in the original PDE the density field has to be differentiable (and thus very smooth because derivatives of the density appear), you can write this PDE in a different equivalent form in which the density field only needs to be integrable (and as such accepts kinks and discontinuities).

You can find examples of the process by googling for weak solutions of the Euler eqts, Navier-Stokes, Maxwell, convection-diffusion-reaction, Stokes eqt...

(*) by multiplying the PDEs with spaces of sufficiently smooth test functions, integrating over the domain, and shifting the derivatives from the unknowns to the test functions by means of the divergence Theorem