Global Spherically Symmetric Solutions For A Coupled Compressible Navier-Stokes/Allen-Cahn System In 3D

In this paper,we consider a coupled compressible Navier-stokes/Allen-Cahn System which described the notion of a mixture of two viscous compressible fluids.We consider the existence and uniqueness of global classical solution,the existence of weak solution and the existence of unique strong solution under the assumption of spherically symmetric condition for initial data in 3D.The main content of the paper is arranged as follows:In Chapter one,we mainly introduce the historical development background and relevant research status of Navier-stokes/Allen-Cahn System,as well as the contents and main results of this paper.In Chapter two,under the hypothesis of spherically symmetric model,the primary equation system can turn into another form which would be simple to investigate with the Euler coordinate transformation and Lagrangian mass co-ordinate transformation methods.By defining suitable function spaces to construct a compression mapping,we proved the local existence and uniqueness of the classical solution with the microlocal analysis theory and the interpolation theorem(Schauder theory etc.).In chapter three,under the assumption of the regularity of the initial value of equations(such as the conservation of mass conservation of energy,etc.),we can first get energy identity.Based on the energy identity and the regularity hypothesis,the result of regularities estimate of the density of fluid ?,velocity fieldu and phase variable ?,which can be given with variety of interpolation theorems and related inequalities(Sobolev's interpolation theorem,holder's inequality young inequality,etc.).In chapter four,based on priori estimates in the third chapter,we can prove the existence and uniqueness of global classical solution with the reduction to absurdity.Reuse the result of the a priori estimate,and the related theorems of the standard mollification and functional analysis,we can concluded that the existence of weak solution and the existence of strong solution.By the Gronwall's inequality we obtain the uniqueness of strong solution.