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

研究了辐射流体动力学中双曲-椭圆耦合系统的非线性波的稳定性及其相关方程。利用傅里叶变换，给出了双曲-椭圆耦合系统线性化系统的基本解的表示公式，并证明了基本解的主部是用热核的形式显式给出的。此外，我们还得到了误差项的精确的逐点估计。利用基本解的表示公式和相应的估计，得到了双曲-椭圆耦合系统解的点向衰减估计。进一步证明了该解对于以相应特征速度传播的扩散波的叠加是渐近的。讨论了双曲-椭圆耦合系统的奇异极限。证明了在此极限下，双曲-椭圆耦合系统的解收敛于相应的双曲-抛物耦合系统的解。我们证明了离散玻尔兹曼方程在半空间中平稳解的存在性。证明了平稳解以指数方式逼近远场，并在大时间内渐近稳定。研究了半空间中等熵Navier-Stokes方程的非线性波的渐近行为。对于流出问题，我们证明了(1)驻波，(2)稀疏波，(3)驻波和稀疏波的叠加非线性波的渐近稳定性。

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.