Research on global solutions in time and their asymptotic behaviors for viscous and relaxation models of conservation laws

1.研究了可压缩流体一维等熵模型半空间上的初边值问题。在流体流入边界的情况下，证明了边界层解与粘性激波（或稀薄波）的叠加在一些较小的条件下是渐近稳定的。在流体在边界处流出的情况下，证明了边界层解的存在性及其渐近稳定性。进一步研究了一维理想气体模型（3 × 3系统）在半空间上的初边值问题，证明了粘性接触波的渐近稳定性。在初始扰动积分为零的情况下，证明了柯西问题的粘性接触波的渐近稳定性。在多维问题中，证明了可压缩Navier-Stokes方程等温模型的球对称解在时间上全局存在并趋于平稳解。研究了描述半导体中电子运动的一维模型的初边值问题。证明了即使对于任何大的掺杂谱线，也存在平稳解，并且是渐近稳定的。对于多维模型，提出了一种高分辨率数值计算迭代算法来模拟稳态解。研究了多孔介质中可压缩粘性流体一维模型解的渐近行为。证明了抛物型方程的解趋向于扩散波，并得到了扩散波渐近的精确衰减率。研究了非线性色散方程和具有临界非线性的耗散方程解的渐近性质。证明了耗散波动方程的解趋向于热方程的自相似解。对于色散方程，给出了线性化方程的特征频率数与非线性项的特征频率数之间的共振现象如何影响解的渐近行为的新结果。少

1.Initial boundary value problems on the half space to one-dimensional isentropic models for compressible fluid are investigated. In a case where the fluid inflows on the boundary, it is proved that the superposition of the boundary layer solution and viscous shock wave (or rarefaction wave) is asymptotically stable under some smallness conditions. In a case where the fluid outflows on the boundary, the existence of the boundary layer solution and its asymptotic stability are also proved. Furthermore, an initial boundary value problem on the half space to a one-dimensional ideal gas model (3 by 3 system) is investigated, and then the asymptotic stability of the viscous contact wave is proved. The asymptotic stability of the viscous contact wave for the Cauchy problem is also proved, if the integral of the initial perturbation is zero. Among multi-dimensional problems, it is proved that the spherically symmetric solution of an isothermal model of the compressible Navier-Stokes equation … More globally exists in time and tends toward its stationary solution.2.Initial boundary value problems for a one-dimensional model which describes the movement of electrons in semiconductors are investigated. It is proved that even for any large doping profile the stationary solution exits and is asymptotically stable. As for the multi-dimensional models, an iterative algorithm of numerical computation with high resolution to simulate the stationary solutions is developed.3.Asymptotic behavior of solutions of a one-dimensional model of compressible viscous fluid in porous media is investigated. It is proved that the solution tends toward a diffusion wave of a parabolic equation, and the precise decay rate of asymptotics to the diffusion wave is also obtained.4.Asymptotic behavior of solutions of nonlinear dispersive equations and dissipative equations with a critical nonlinearity is investigated. It is proved that the solution of dissipative wave equation tends toward a self-similar solution of a heat equation. As for the dispersive equations, a new result on how the resonance phenomena between the characteristic frequency number of the linearized equation and that of nonlinear term influences the asymptotic behavior of the solutions. Less