Asymptotic Behavior For Euler-Poisson Equations And Euler Equations

In this thesis, we first summarize the hierarchy and recent development of semi-conductor device modeling, and study the perturbation problem near equilibrium for themulti-dimensional Euler-Poisson equations (the classical hydrodynamic model) in semi-conductors and the multi-dimensional compressible Euler equations with damping. Weapply some techniques based on Littlewood-Paley decomposition and paradifferential cal-culus to lower the space regularity of the initial data, and improve some known resultsin Sobolev space. We prove the global existence, uniqueness and stability of classicalsolutions on the framework of Besov space. Moreover, on such framework, the relatedrelaxation-time limit problem is also investigated.More specifically speaking, we study three problems, which are given in Chapter 3,Chapter 4 and Chapter 5 respectively.In Chapter 3, as the first step, we consider the Cauchy problem for the simplified(isentropic and isothermal case) hydrodynamic model (1.2.2) for semiconductors in themulti-dimensional space. Provided that the initial data is a disturbance close to equilib-rium, we establish the global existence, uniqueness and exponential stability of classicalsolutions to the hydrodynamic model. In the past ten years, like the well-posedness andstability of smooth solutions, many different authors have made great efforts on the Cauchyproblem and the initial boundary problem, in the one-dimensional or multi-dimensionalspace. Those results are obtained on the framework of of Sobolev space H^l(R^d) and theregularity index is required to be high (l＞1+d/2,l∈Z) when one deals with them byclassical analysis methods. In this chapter, we shall investigate the limit case l=1+d/2where Kato's classical existence theory fails, the critical space achieved is a class of Besovspaces B_2,1^1+d/2(R^d) rather than H^1+d/2(R^d). First, we prove a local existence and unique-ness result of classical solutions for the general initial data in terms of regularized meansand the compactness argument. Although it is proved via a hyperbolic symmetric system,some especial contents from Poisson equation are to be dealt with. Second, we establisha global existence result with exponential decay for small initial data. The used methodis the high- and low-frequency decomposition one. As we seen, Having relaxation doesmatter only for the global existence result, on the other hand, the Poisson equation plays akey role in the estimate of low frequency, which leads to the exponential decay of classicalsolutions. Here, we do not give any geometrical assumptions of solutions on the gen-eral multi-dimensional unbounded domain. Moreover, Based on the global existence ofclassical solutions, we characterize the exponential decay of the vorticity in Besov space. Following the similar process, we extend these results to the full hydrodynamic model(1.2.1) with zero thermal conductivity. Besides more tedious calculations, some new trou-bles will occur in the spectral localization estimates. Fortunately, the information behindthe mass and momentum equations help us eventually to eliminate them. For brevity, weonly state the related results for the full hydrodynamic model for semiconductors, for de-tails, see [24]. These results are regarded as the improvement of Hsiao's et al. [38] andAli's [1].In Chapter 4, the Cauchy problem for the compressible Euler equations with dampingis studied on the framework of Besov space. By contrast with the simplified hydrodynamicmodel for semiconductors, there is a lack of coupled Poisson equation for electric poten-tial. We still consider the perturbation near equilibrium. Based on the effort on Euler-Poisson equations in Chapter 3, the local existence and uniqueness of classical solutionsto the compressible Euler equations with damping can be obtained similarly, and the resultincludes the one-dimensional case. Due to the difference of low-frequency estimates, weneed to choose a more stronger space B_2,2^1+d/2+ε(R^d)(ε＞0) than B_2,1^1+d/2(R^d) to obtain theglobal existence of classical solutions to the compressible Euler equations with damping.Due to the techniques, we restrict the space dimension to d≥3. As a direct consequence,we show the large-time asymptotic behavior of classical solutions in Besov space. Finally,we characterize the exponential decay of the vorticity, too. These results are viewed as theimprovement of Sideris's et al. [75].In Chapter 5, we shall observe an interesting phenomenon by a large time scale withrespect to the relaxation timeτ. For the isothermal Euler-Poisson equations for semiconductors, we refine the a-priori estimates in Chapter 3 such that the positive constantsare independent ofτin the global existence theorem. Then, using the weak convergencemethod and compactness argument, the relaxation-time limit (τ→0) can be performed.The scaled solutions are shown to converge towards that of the drift-diffusion model(5.1.6) for semiconductors. As a by-product, the global existence of weak solutions tothe multidimensional drift-diffusion model is obtained. For the isotropic Euler-Poissonequations, we can establish the corresponding relaxation-time limit result in Besov spaceB_2,2^1+d/2+ε(R^d)(ε＞0). Following the similar process, we also investigate the relaxation-time limit for the compressible (isentropic or isothermal) Euler equations with dampingand prove that the density converges towards the solution to the porous media equation(5.2.7). For the thermal case, we give another proof, however, the relaxation limit result iscoincident.