Analysis Of Solutions For Viscous Fourth-order Nonlinear Parabolic Equations

The fourth-order parabolic equation is often used to describe and analyze the thin viscous incompressible flow along the inclined plane,or simulate the fluid flow,such as the foam thin layer analysis and the movement of the tear fluid under the action of the contact lens.In this paper,the existence of weak solution of one-dimensional thin film equation with initial boundary condition is studied.Moreover,in high dimensional space,the existence and uniqueness of weak solutions for a class of viscous fourth-order parabolic equations are studied.The main research questions are as follows:1.The existence of solution of viscous film equation in one dimensional space:u where T>0,m(u)=u,?=(-1,1),Qr=?×(0,T)and ?=(?)?×(0,T).This model can be regarded as a Cahn-Hilliard equation with degeneration mobility.In this paper,the entropy functional method is used to overcome the difficulties caused by degeneration mobility and viscous terms,and then we obtain the existence of non-negative weak solutions.By constructing the appropriate approximation equation and entropy functional,we obtain a uniform estimation which is not related to the approximation parameter.The approximate solution satisfies the uniform estimation.By obtaining the limit of the small parameters,the existence of the weak solution is obtained.2.The existence and uniqueness of weak solutions for a class of viscous fourth-order parabolic equations in high dimensional space.where ?(?)RN is a bounded open area that is smooth enough,and v0(x)is an initial function.The constant p>1,k,?>0,and k(?)?v/(?)t is a viscous relaxation factor or a viscous effect.With the condition of the initial boundary value,the existence problem of the solution of semi-discrete elliptic equations is constructed by time discretization.Combining the Poincare inequality and the Young inequality,the existence of discrete problems is obtained.The approximation solution of this equation is constructed by using Galerkin method,and then the uniform estimation is obtained.Convergence of the approximation solution gives the existence of weak solution.