Three Dimensional Viscous Liquid - Gas Two Phase Flow Model Of Vacuum Problem The Existence Of Global Classical Solution

This paper deals with the3D viscous liquid-gas tow phase flow model in the following form Where the variables m=αlρl, n=αgρg, u={u1,u2,u3) and P=P(m,n) denote the liquid mass, gas mass, the velocity of the liquid and gas, the common pressure for both fluids, respectively; μ and λ are viscosity constants, satisfying μ>0,2μ+3λ≥0. The other unknown variables ρl and ρg denote liquid and gas densities, αl, αg∈[0,1] denote the liquid and gas volume fractions, satisfying αl+αg=1.The model0.2can be regarded as the simplified form of the model, which are widely used in the oil industry to describe the production and transport of the gas and oil throng long well or pipelines. In this paper, we consider the viscous effect, neglected the effect of gas in mixture momentum equation and the external force.We mainly study the Cauchy problem about model0.2. We obtain the global existence of the classical solutions when the initial condition with vacuum under the small assumptions on the initial energy norm. The embedding theo-rems in the1D are different from the3D case. It is difficulty to prove the upper bound of m and n directly. In order to get the pointwise bounds of m and n and high order estimate of u, we need some priori assumptions. Here we apply the smallness hypothesis on the initial energy to obtain assumptions of rationality. Then we use the classical continuation method to close the priori assumptions. Finally, we obtain the upper bound of m and n and high order estimate of u under the small assumptions on the initial energy norm.