Weak solution theories are in general necessary for interface evolution problems as topology changes naturally occur. If the topology change is realized through a physically unstable singularity, this results in non-uniqueness of solutions afterward. The best one can thus expect is a weak-strong uniqueness principle; and this was proven in recent years for prominent examples. At the level of a weak solution concept, the key conceptual ingredient for these results is given by the dissipative nature of the problems. In this talk, I will discuss recently established weak solution theories for two basic examples: mean curvature flow and Mullins-Sekerka flow. Both solution theories are essentially only encoded in terms of a single sharp energy dissipation principle, taking direct inspiration from De Giorgi's approach to gradient flows or the Sandier-Serfaty approach to evolutionary Gamma-convergence. This is joint work with Tim Laux and Kerrek Stinson.