From hyperbolic to parabolic PDEs by quadratic change of time with applications to fluid mechanics and geometry

We show how several parabolic equations and systems can be derived from hyperbolic systems of conservation laws just by performing a suitable quadratic change of time. The simplest example is the heat equation which follows from the isothermal Euler equations. This idea allows us to transfer well known techniques from the hyperbolic world to the parabolic setting, for instance for some mean-curvature flows.