This talk will present recent works to identify the critical scales at which regularity is propagated by advection equations with rough, i.e. non-smooth, velocity fields. After reviewing the classical theory of renormalized solutions which provides qualitative arguments of regularity and well-posedness, more recent quantitative approaches will be discussed. Our goal is to use this framework on complex systems where the advection equation is coupled to other PDE’s. A driving example is the compressible Navier-Stokes system for realistic state laws and in particular for non-monotone pressure laws that are thermodynamically unstable.This naturally leads to focus on advection equations with compressible flows for which we find a log log scale of propagation. The corresponding main result for the compressible Navier-Stokes system is a joint work with D. Bresch.