The Global Existence Of Weak Solutionsand The Large-time Behavior For A Class Of Non-Newtonian Fluid Dynamic Equations

The fluid-particle coupling system is one of the research hot-spots in the field of partial differential equation fluid mechanics,which describes the micro-scopic motion of dispersed particles in fluids and the viscosity law of fluids.The particle equation and the Navier-Stokes fluid coupling equations has been quite rich results,mainly to study the well-posedness problem and the nature of the solution.Non-Newtonian fluid refers to a fluid that does not satisfy Newton's law of viscosity,that is,a fluid with a nonlinear relationship between the stress tensor and the strain rate tensor.At present,there are not many literatures about the particle equation and non-Newtonian fluids coupling equations.This article mainly considers the Vlasov-Fokker-Planck equation and a class of com-pressible non-Newtonian fluid coupling equations,the Vlasov equation and an incompressible non-Newtonian fluid coupling equationsOn the one hand,the literature[10]gives the global existence of weak solu-tions for a class of compressible non-Newtonian fluid equations.Referring to the research methods in the literature[25]this paper establishes the Vlasov-Fokker-Planck equation and a class of compressible non-Newtonian fluids coupling e-quations in a three-dimensional bounded domain.Under the condition that the initial density of the fluid has upper and lower bounds,the divergence of the fluid velocity is bounded,the effect of the friction between the fluid and the particles is overcome by truncating the fluid velocity,the fixed point theorem and weak convergence method are used to prove the global existence of the weak solution for the coupling equationsOn the other hand,the literature[27]gives the global existence of weak solu-tions for the Cucker-Smale equation and a class of incompressible non-Newtonian fluid coupling equations.In this paper,the Vlasov equation and a class of in-compressible non-Newtonian fluids coupling equations is established.The global existence of the weak solution can be similarly derived by the literature[27].Un-der the condition of small initial energy,we employ a Lyapunov functional to estimate its time decay rate,the energy of the weak solution of particles and fluids decreases exponentially with time.