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The main purpose is to generalize a theorem of Arendt about uniqueness of$C_0$-semigroups from Banach space setting to the general locally convex vectorspaces, more precisely, we show that cores are the only domains of uniquenessfor $C_0$-semigroups on locally convex spaces. As an application, we find anecessary and sufficient condition for that the mass transport equation has oneunique $L^1(\R^d,dx)$ weak solution.

Uniqueness of a pre-generator for $C_0$-semigroup on a general locally convex vector space