Study On The Global Well-Posedness And Long Time Behavior Of Solutions For Euler-Poisson Equations

In this paper,we study the whole well-posedness of Euler-Poisson equa-tions and the long time behavior.As an important part of the fluid dynamics model,the Euler-Poisson equations obtained more and more attention from mathematics,physics and biology,it describes the physical now including semi-conductor devices in the transport of electrons and holes,plasma of cationic and anionic transmission,the flow of the gas giants internal particles and particle transport channel proteins in biology and so on.In this paper,we consider the Euler-Poisson system as follows:where ? is a smooth bounded domain in RN(N=1,2,3),and ?1??2,u1,u2?????denote the electron density,the hole density,the electron velocity,the hole ve-locity,the potential function and the electric field respectively.?1>0 and?2>0 are the constants for the relaxation time of the momentum for electrons and ions.?>0 is the Debye length.D(x)>0 is the doping profile and smooth enough.P(?i)and P(?2)represent the pressure of the electrons and the holes,respectively.For the isentropic flow the pressure functions are where ? is the adiabatic exponent.Now we consider the system(0.4)with insulating boundary conditions where v is the unit outward normal vector along(?)?.The initial conditions are prescribed as ?i(x,0)>0 and ui(x,0),and satisfies the compatibility condi-tion ui(x,0)·?|(?)?=0.The equations derived from the fluid dynamics model of semiconductor.The mathematical model theory or the partial differential equation method of semiconductor and superlattice is one of the important research topics in the modern semiconductor industry and the application of maths.Theory and numerical research for model of semiconductor associat-ed with many branches of mathematical physics disciplines,such as quantum mechanics and statistical mechanics,partial differential equations,functional analysis,random analysis,geometric measure theory,etc.At the same time,with the generalization of miniaturization of the semiconductor industry and the nanometer technology,it has become a challenging mainstream research direction of international applied mathematics.Therefore,The research on the behavior of solutions to Euler-Poisson equations is not only of scientific signif-icance,but also of certain application value.The main contents of this paper are as follows:In chapter one,we introduce the background of the research and the main work of this paper.In chapter two,we give the preliminary knowledge,and introduce the math-ematical terms and tools being used in this paper.In the third chapter,we studied the long time behavior for the solutions of the above system(0.4).Combining with the Poisson equation in the system(0.4),we inspire by[19]to use a symmetrizer to reduce the system(0.4)to a symmetric hyperbolic one in the sense of Friedrichs.And then apply the basic energy estimate to investigate that the long time behavior for the steady solutions.Of course,due to the mutual coupling of the electrons and holes,the method of[19]cannot be directly popularized and applied.To solve this problem,we refer to the method in[25]and introduce a new form of electric field equation based on system(0.4).This method enables the energy estimate to be carried out effectively,so as to obtain the low-order to high-order energy estimate of the global smooth solution.The fourth chapter,this chapter skillfully using the variational method and the maximum and minimum principles to get the existence of steady-state solutions for the system(0.4)with the isentropic and the isothermic case,re-spectively,and basing on the embedding theorem,a prior estimate and Schauder estimate to improve the regularity of this solution,and then the corresponding smooth solution is obtained.To our knowledge,this is the first existence re-sult of non-constant steady smooth solution with non-flat doping profile for the multi-dimensional isentropic bipolar model.