Global Existence Of Solutions To The Compressible Navier-Stokes-Smoluchowski Queations

Navier-Stokes equations are used to describe viscous Newtonian fluid in hydrodynamics,Navier-Stokes equations are also used to describe the conservation of momentum of viscous in-compressible fluid.Navier-Stokes equations are the basic equations in partial differential equa-tion.The main contents of this paper are concerned with Cauchy 's problem of the Navier-Stokes equations coupled with Smoluchowski equation for compressible ideal gas in three-dimensional space.In this paper,we mainly consider the initial value problem of the system of equations in a bounded smooth region of R3.By perturbing the initial value of the stationary solution in some Sobolve space,using the local existence theorem of the solution in the existing fluid-structure coupling model and using the classical energy estimation method,a series of prior estimates independent of time T are established for the strong solution,so as to obtain the theory of glob-al existence of the solution in H3 space.Fluid-solid coupling model play an important role in the settlement analysis and treatment of dispersed suspension of fluid particles.It has attract-ed wide attention in haze,sewage treatment,combustion of active liquid formation and other phenomena.The full text is divided into three chapters.Chapter one mainly introduces the basic com-pressible Navier-Stokes equations of viscous fluid,compressible Navier-Stokes-Smoluchowski equations and the optimal state of Navier-Stokes-Smoluchowski equations.The research status of hydrodynamic equations at home and abroad and the difficult conjectures about hydrody-namic equations at present are also introduced.The last part of the first chapter gives the main research methods and conclusions of this paper.The second chapter of this paper is mainly about the basic knowledge,listing the basic inequalities and the related proofs of inequalities.Chapter three is the proof of Theorem 1.3.The first part of the third chapter is to analyze and transform the main models of the paper to obtain the stationary solution and strong solution.The second part is to estimate the strong solution priorily and obtain the low-order and high-order estimates of the strong solution by energy estimation method,so as to obtain the global existence theory of the strong solution to be proved in this paper.