Existence Of Global Weak Solutions And Steady State Solutions Of Time-dependent Thomas-fermi Equations

With the establishment of quantum mechanics,the application of quantum mechanics to various fields has become a key problem,especially in the understanding of the micro world.Schr¨odinger equation is the basic equation of(non relativistic)quantum mechanics and its position is equivalent to that of Newton equation in classical mechanics.The time-dependent Thomas-Fermi equation is obtained by Madelung transformation of Schr¨odinger equation.The equation describes the macroscopic dynamic equation of the identical particle system whose coordinate origin is taken on the atomic nucleus and in the external potential field formed by the atomic nucleus.This paper mainly studies the existence of global weak solutions,steady-state solutions and nonlinear stability of time-dependent Thomas-Fermi equations.The paper is divided into four chapters:In Chapter 1,we introduce the research significance,background,research status and main research conclusions of time-dependent Thomas Fermi equations.In Chapter 2,we introduce some basic inequalities,lemmas,theorems,corollaries and so on.In Chapter 3,we mainly study the existence of weak solutions for onedimensional time-dependent Thomas-Fermi equations.Compared with the onedimensional semiconductor Euler-Poisson equations,the equations have one singular point,that is,the derivative is infinite at the origin.Although the equation contains a singularity,we can establish the uniform estimation of the solution of one-dimensional time-dependent Thomas-Fermi equations through observation and calculation.Therefore,we can obtain the existence of global weak solutions of one-dimensional time-dependent Thomas Fermi equations by using the method of finding global weak solutions of one-dimensional semiconductor Euler Poisson equations.Firstly,the approximate solution equation is constructed by viscosity vanishing method.Secondly,the existence and uniform estimation of approximate solutions are obtained by using Poisson formula and maximum principle.Finally,using the compensated compactness method,the approximate solution converges almost everywhere in the sense of distribution,so that there are weak solutions for one-dimensional time-dependent Thomas-Fermi equations.This also provides a theoretical basis for the existence of spherically symmetric solutions of three-dimensional time-dependent Thomas-Fermi equations.In Chapter 4,we mainly study the existence and nonlinear stability of steadystate solutions of three-dimensional time-dependent Thomas-Fermi equations.Based on Lieb’s proof that three-dimensional time-dependent Thomas Fermi equations have a set of steady-state solutions with zero velocity,we prove that threedimensional time-dependent Thomas Fermi equations have a set of steady-state solutions with non-zero velocity.The proof method is to prove the existence of steady-state solutions by using the variational method in elliptic equations.Then,according to the second law of thermodynamics,the energy of three-dimensional time-dependent Thomas-Fermi equations is non increasing with respect to time.Finally,it is proved that the energy is stable.