Equivalence of weak solution concepts for mean curvature flow

We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which include Ilmanen’s boundary motion as well as combined Brakke flows and distributional solutions. We prove that if one has a \enquote{foliation} by variational solutions, then the resulting function is the unique viscosity solution. This answers an open question suggested by the work of Evans and Spruck [J. Geom. Anal. 5 (1995), pp. 77–114], Ilmanen [Mem. Amer. Math. Soc. 108 (1994), pp. x+90], and the authors [J. Geom. Anal. 34 (2024), p. 18]. These results show that almost every level set of the viscosity solution is a variational solution. Thus, we establish the equivalence of these solution concepts. Moreover, we show the generic uniqueness of variational solutions.