Boundedness And Lager Time Behavior Of Weak Entropy Solutions To The Euler Equations With A Source Term

In this paper,we first consider the uniform boundedness and large time behavior of weak entropy solutions to the Cauchy problem and the initial-boundary value problem of a compressible Euler equation with a specific source term:(?) where x ∈I,I=R(or I=[0,1]),t ∈[0,T)denote the space variable and the time variable respectively.Then we study the global existence and large time behavior of weak entropy solutions to the unipolar hydrodynamic model for semiconductor devices:(?) where x ∈ R,t ∈[0,T)denote the space variable and the time variable respectively,too.This article is divided into four chapters.In chapter one,we firstly introduce the generation and development of hydrodynamics as a subject.Then the physical background and research significance of the compressible Euler equation with a specific source term and the unipolar hydrodynamic model for semiconductor de-vices are introduced respectively.Finally,we summarize the related studies of the two models at home and abroad,and explain the primary contents in this paper.In chapter two,we first give sufficient conditions of uniform boundedness for any weak entropy solution,which solves the Cauchy problem of Euler equation with a spe-cific source term.Then,we obtain the global existence for weak entropy solutions to both Cauchy problem and initial-boundary value problem by virtues of Lax-Friedriches scheme and compensated compactness method,and provide sufficient conditions of uniform boundedness in time for the above weak entropy solution.In chapter three,we study the global existence,as well as the rate of growth with time of weak entropy solutions obtained by the viscosity vanishing method to the unipolar hydrodynamic model for semiconductor devices by using the improved maximum principle.In chapter four,large time behavior of weak entropy solutions to the two models are concerned.Via energy method and entropy estimation,the global weak entropy solutions of the Euler equation with a specific source term are proved to converge exponentially to the steady state solutions in L2.At the same time,boundedness and exponential convergence of weak entropy solutions to the unipolar hydrodynamic model for semiconductor devices obtained by viscosity vanishing method when the adiabatic index γ>3.