Global Weak Solutions to a Fluid-particle System of an Incompressible Non-Newtonian Fluid and the Vlasov Equation

The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation. The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for \(p\geqslant {11\over 5}\) $$p⩾\frac{11}{5}$$ . The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function. The advantage of the new formulation is to control the external force term \(G=-\int_\mathbb{{R}^{d}}(\mathbf{u}-\mathbf{v})fd\mathbf{v}\ (d=2,3)\) $$G=-{\int }_{{\mathbb{R}}^{\mathbb{d}}}(\mathbf{u}-\mathbf{v})fd\mathbf{v}(d=2,3)$$ . The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques. We further prove the uniqueness of weak solutions to the considered system.