Mild solutions are weak solutions in a class of (non)linear measure-valued evolution equations on a bounded domain

Abstract

We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain $[0,1]$. Mass moves, driven by a velocity field that is either a function of the spatial variable only, $v=v(x)$, or depends on the solution $\mu$ itself: $v=v[\mu](x)$. The flow is stopped at the boundaries of $[0,1]$, while mass is gated away by a certain right-hand side. In previous works [J. Differential Equations, 259 (2015), pp. 1068-1097] and [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953], we showed the existence and uniqueness of appropriately defined mild solutions for $v=v(x)$ and $v=v[\mu](x)$, respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when $v=v(x)$ and when $v=v[\mu](x)$.