About the concept of measure-valued solutions to distributed parameter systems

The measure-valued (mv) solutions are commonly defined in the literature simply by putting Young measures into the respective partial differential equations. Here a few examples of mv-solutions to evolution problems (conservation laws, fluid dynamics and a so-called backward-forward heat problem) are re-investigated to show that such definitions are not satisfactorily selective, i.e. they admit (sometimes surprisingly) many mv-solutions apparently without any physical sense. Then an attempt for a selective definition in the case of the backward-forward heat problem is made, resulting to a certain evolution variational inequality.