Study on the fundamental solutions to the equations of radiating gases and its applications

辐射气体方程基本解的研究及其应用

基本信息

批准号：
11440049
负责人：
KAWASHIMA Shuichi
金额：
$ 6.53万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

We study the stability of nonlinear waves for hyperbolic-elliptic coupled systems in radiation hydrodynamics and related equations.1. By using the Fourier transform, we give a representation formula for the fundamental solutions to the linearized systems of hyperbolic-elliptic coupled systems and verify that the principal part of the fundamental solutions is given explicitly in terms of the heat kernel. Also, we obtain the sharp pointwise estimates for the error terms.2. We obtain the pointwise decay estimate of solutions to the hyperbolic-elliptic coupled systems by using the representation formula for the fundamental solution and the corresponding estimates. Furthermore, we prove that the solution is asymptotic to the superposition of diffusion waves which propagate with the corresponding characteristic speeds.3. We discuss a singular limit of the hyperbolic-elliptic coupled systems. We prove that at this limit, the solution of the hyperbolic-elliptic coupled system converges to that of the corresponding hyperbolic-parabolic coupled system.4. We show the existence of stationary solutions to the discrete Boltzmann equation in the half space. It is proved that the stationary solution approaches the far field exponentially and is asymptotically stable for large time.5. We study the asymptotic behavior of nonlinear waves for the isentropic Navier-Stokes equation in the half space. For the out-flow problem, we prove the asymptotic stability of nonlinear waves such as (1)stationary wave, (2)rarefaction wave, and (3)superposition of stationary wave and rarefaction wave.

我们研究了辐射流体力学中双曲椭圆耦合系统非线性波的稳定性及相关方程： 1．通过使用傅里叶变换，我们给出了双曲椭圆耦合系统线性化系统基本解的表示公式，并验证了基本解的主要部分是根据热核明确给出的。此外，我们还获得了误差项的尖锐逐点估计。2．我们通过使用基本解的表示公式和相应的估计来获得双曲椭圆耦合系统解的逐点衰减估计。进一步证明了该解渐近于以相应特征速度传播的扩散波的叠加。 3．我们讨论双曲椭圆耦合系统的奇异极限。我们证明了在该极限下，双曲-椭圆耦合系统的解收敛于相应的双曲-抛物线耦合系统的解。 4．我们证明了半空间中离散玻尔兹曼方程平稳解的存在性。证明了平稳解以指数方式逼近远场，并且在长时间内渐近稳定。 5．我们研究半空间中等熵纳维-斯托克斯方程的非线性波的渐近行为。对于流出问题，我们证明了非线性波的渐近稳定性，例如（1）驻波，（2）稀疏波，以及（3）驻波和稀疏波的叠加。