Global Weak Solution To The System Of Compressible Non-Newtonian Fluid Coupled With Vlasov-Fokker-Planck Equation

In this paper,we investigate the initial-boundary value problem to a Vlasov-Fokker-Planck equation coupled with a compressible non-Newtonian fluid in a bounded domain of three-dimensional,and the initial density of the system is nonnegative.The existence of global weak solution is established by an ener-gy method and the monotone operator theory.In particular,the key point of our main result is to deal with the nonlinear terms of the compressible non-Newtonian fluid for the case p?3.The content of this paper is arranged as follows.First of all,the initial boundary value problem of the coupling system of kinetic and fluid equations is established by referring to the model of the reference[18,26].Secondly,assuming that the initial value has finite energy,the energy es-timation of the system is obtained by using the basic energy method,and the definition of the weak solution of the system is given.Then,we construct a suitable approximation by truncating the velocity of the fluid.The existence of the approximation solution is obtained by using the fixed point argument,and the uniform estimation of the approximation solution is made.Finally,using the above uniform estimate,we prove the compactness of the approximate solution.The existence of the global weak solution of the origi-nal coupled model is obtained by taking the limit of the approximate solution.