We propose a universal, conceptually simple, and systematic method to prove well-posedness of Cauchy problems for weak solutions of linear parabolic equations with a non-smooth, time-dependent elliptic part, which has a variational definition. Our classes of weak solutions are considered with minimal assumptions. 

We prove the existence and uniqueness of a fundamental solution, which seems to be new in this generality. One of our main tools is embedding results, which ensure the time continuity of our weak solutions, going beyond the celebrated Lions regularity theorem and addressing a variety of source terms. We conclude this presentation with some concrete applications. 

This talk is based on a joint work with Pascal Auscher.