Measure-valued evolution equations with solution-dependent velocities on bounded domains: Mild solutions are weak solutions

Joep H. M. Evers

Published 2016-06-04Version 1

In [J. Differential Equations, 259 (2015), pp. 1068-1097], we proved the well-posedness of a measure-valued evolution equation posed on the bounded domain $[0,1]$. This result was obtained in the framework of (measure-valued) mild solutions; that is, solutions to the variation of constants formula. We were restricted there to a prescribed velocity field $v:[0,1]\to\mathbb{R}$ that is only a function of the spatial variable. We generalized this result in [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953] to velocity fields that depend on the solution itself: $v=v[\mu](x)$. To that aim, we first constructed so-called Euler approximations. These are approximate solutions, per time interval $(t_j,t_{j+1}]$ driven by an appropriate $v_j=v_j(x)$. In [SIAM J. Math. Anal., 48 (2016), pp. 1929-1953], mild solutions were defined as the limit of a sequence of Euler approximations as the mesh size went to zero. Such mild solutions were showed to exist and be unique. In the current paper we define weak solutions (for $v=v(x)$ and $v=v[\mu]$, respectively) and show that, in either case, the aforementioned mild solutions are weak solutions.