Weak Solutions for a non-Newtonian Stokes-Transport System

In this article, we study a non-Newtonian Stokes-Transport system. This set of PDEs was introduced as a model for describing the behavior of a cloud of particles in suspension in a Stokes fluid, and is a nonlinear coupling between a hyperbolic equation (Transport) and a nonlinear elliptic equation (non-Newtonian Stokes), and as such can be considered as an active scalar equation. We prove the existence of global weak solutions with initial data in critical Lebesgue spaces. In order to overcome the difficulties introduced by the highly nonlinear aspect of this problem, we resort to a combination of DiPerna-Lions theory of transport equations and Minty's trick for elliptic equations.