The Existence Of Solutions To Nonhomogeneous And Nonlinear Boundary Value Problem For Equations Related To NSE

The existence of global smooth solution of three-dimensional Navier-Stokesequation and Euler equation is a open problem. Even existence of strong Lax pair ofEuler equation in 2D is still open. In this paper we mainly investigate the existence ofsolutions to nonhomogeneous and nonlinear boundary value problem for equationsrelated to Navier-Stokes equation which is one of the Clay Institute’s MillenniumProblems, including Euler equation, Schr?dinger equation. The(I, J) similar solutionmethod is presented and some smooth solutions to 2D Euler equation are given. Byusing these result we build the well-posedness of Euler equation. If Navier-Stokesequation and Euler equation are transformed to Schr?dinger equation, we will get lotsof interesting results for rich literature about Schr?dinger equation. Due to Eulerequation can be transformed into complex Schr?dinger equation, so for other cases ofEuler equation, we consider the equations containing fractional order respect to time,fractional order to space and the nonlinear term with fractional order. To all myknowledge, no more than two kinds of situations is studied in the other literature sofar. To solve our questions, we put forward a new kind of weak solution.Section one, we mainly introduce the physical backgrounds and themathematical research situation of those equations related to NS equation.Section two, we mainly investigate the well-posedness of Euler equation. we putforward the(I, J) similar method for incompressible two dimensional Euler equations,and obtain a series of explicit(I, J) similar solutions to the incompressible twodimensional Euler equations. These solutions include all of twin wavesolutions, somenew singularity solutions and some global smooth solutions with the finite energy.We also discover that twin wave solution and affine solution to two dimensionalincompressible Euler equations are respectively plane wave and constant vector. Weprove that the initial boundary value problem of incompressible two dimensionalEuler equations admits unique solution and discuss the stability of the solution.Finally, we supply some explicit piecewise smooth solutions to incompressible threedimensional Euler and an example to incompressible three dimensional Navier-Stokesequations which indicates that viscosity limit of a solution to Navier-Stokes equationsdoes not need to be a solution to Euler equations.Section three, we can study Navier-Stokes equations by Schr?dinger equation forNavier-Stokes equation and Euler equation can be approximately transformed toSchr?dinger equation. In this paper, we consider the equations containing fractionalorder respect to time, fractional order to space and the nonlinear term with fractionalorder. So, in this Section, we probe the existence of weak solutions and attractor ofSchr?dinger equation. Firstly, we define the weak solutions of the Schr?dingerequation of fractional order. Secondly, we give some priori estimates to the equation.At last, we draw some conclusions to our equations. In this Section, we usecompactness theory, Galerkin method and so on.