Research On Moser Iterative Method In Partial Differential Equations

In this paper,we considered the the regularity of weak solutions of the linear elliptic equations,the quasilinear parabolic systems which is a generalization of p-Laplacian of the type and the compressible Navier-Stokes-Poisson equations by Moser iterative method,respectively.In Chapter 1,we briefly introduced the development history of Partial Differential Equations,and we introduced the Moser iterative method from the concept of "weak solution".Then we make a simple summary of the development history of the Moser iterative method.In Chapter 2,we summarized some symbols about mathematical that are often used in the following chapters.And we briefly described some common basic knowledge of Partial Differential Equations,such as common definitions,properties,lemmas and theorems.In addition,some lemmas are proved simply,for example,the related knowledge of Calderó n-zygmund theory is introduced and sorted out in detail.In Chapter 3,we considered the partial regularity of weak solutions to elliptic equations with measurable and bounded divergence coefficients.In the first part of this chapter,we introduced the definition of the weak subsolution(supersolution)of the equation and some properties of the weak subsolution(supersolution),which are used to prove the integrability of the solution through inequality techniques such as Poincare's inequality,Holder's inequality and interpolation techniques in Sobolev space.That is to say,u? Lp0 for some positive constants p0.Finally,we use the Moser iteration method to make the integrability to improve to L?,thus the local boundedness of the weak solution is proved.In addition,in the second part of this chapter,we use the theorem of the first part and embedding theorems to study the regularity of weak solutions of Poisson equations by constructing a new form of weak subsolution(supersolution).In Chapter 4,we considered the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p-Laplacian.And the main part of the equations satisfies some ellipticity and the inhomogeneity term satisfies different growth conditions.We considered the boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and the nonlinear iteration procedure of the Moser type in this paper.Firstly,we studied the regularity of weak solutions and gradient solutions of quasilinear parabolic equations by energy method,different inequalities and Moser iteration method,and obtained the local boundness estimation of corresponding weak solutions.Then,we adjusted the main part of the ellipticity and the inhomogeneity term of the growth condition to considered the the weak solution and gradient solution to the solution of the parabolic equations of which the inhomogeneity term is replaced by natural growth condition by means of energy method,the embedding theorem and Moser iteration methodthe,and obtained the corresponding estimate of the weak solutions and the gradient solutions.In Chapter 5,we considered the regularity of the weak solutions to the compressible Navier-Stokes-Poisson equations of the period domain in dimension three.Firstly,we use the energy method,Leray projection operator,Lagrangian coordinates,Eulerian coordinates,and other theoretical knowledge to study the prior estimation problem of density p;Then,we utilized the integrability condition of velocity u to obtain the integrability of velocity u;Last,we took advantage of the Moser iteration method to prove the L?estimate of u.In Chapter 6,we made a summary of the Moser iteration method.