Vlasov equation coupled with non-Newtonian fluids with discontinuous-in-time stress tensor

We analyze the system of equations describing the flow of a dilute particle system coupled with an incompressible non-Newtonian fluid in a bounded domain. In this setting, both PDEs are connected via a drag force, or the friction force. We are interested in a thin spray regime, which means that we neglect the inter-particle interactions. When it comes to the fluid system, the Cauchy stress tensor is supposed to be a monotone mapping and has asymptotically $(s-1)$-growth with the parameter $s$ depending on the spatial and time variable. We do not assume any smoothness of $s$ with respect to time variable and assume the log-Hölder continuity with respect to spatial variable. An example of materials, which satisfy those assumptions, are those whose properties are instantaneous, e.g. changed by the switched electric field. We show the long time and the large data existence of weak solution provided that $s\ge\frac{3d+2}{d+2}$.