In the answer to https://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution

H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions", Funkcial Ekvac. Ser. Int. 38 (1) (1995) 101–120.(pdf)

is mentioned. This holds in the elliptic case. Do we know anything about parabolic equations?