Study on the initial value problem for quasilinear hyperbolic-elliptic coupled systems

研究了一类双曲-椭圆型耦合方程组的初值问题，其中包含一个辐射气体方程。对于最简单的辐射气体模型系统：1.给出了经典整体解不存在和存在的充分条件；2.证明了Sobolev空间中光滑解的整体存在性和渐近衰减性。该解逼近由粘性Burgers型方程的自相似解定义的扩散波。3.证明了由无粘性Brugers型方程的中心稀疏波定义的稀疏波的渐近稳定性。4.证明了激波剖面行波解的存在性。只有当冲击强度大于临界值时，这些冲击波才具有不连续性。此外，我们还证明了光滑激波的渐近稳定性。5.我们证明了黎曼问题弱解的对数存在性。弱解中包含的跳跃是指数衰减的，随着时间趋于无穷大，解趋于相应的光滑激波。6.给出了熵函数的数学定义，并证明了系统的对称化与熵函数存在性的等价性。其次，我们给出了稳定性条件，并在此条件下证明了Sobolev空间中光滑解的整体存在性和渐近衰减性。此外，我们还观察到相应的双曲-抛物耦合方程组的解是时间渐近逼近的。这些结果适用于辐射气体的方程。

We study the initial value problem for a class of hyperbollic-elliptic coupled systems including the equation of a radating gas. For the simplest model system of a radiating gas :1.We give sufficient conditions for the non-existence and the existence of classical global-solutions.2.We prove the global existence and asymptotic decay of smooth solutions in Sobolev spaces. The solution approaches the diffusion wave which is defined in terms of the self-similar solution of the viscous Burgers equation.3.We prove the asymptotic stability of rarefaction waves which are defined in terms of the centered rarefaction wave of the inviscid Brugers equation.4.We show the existence of traveling wave solutions of shock profile. These shock waves have a discontinuity only when the shock strength is greater than a critical value. Moreover, we prove the asymptotic stability of smooth shock waves.5.We show the glogal existence of weak solutions to the Riemann problem. The jump contained in the weak solution decays exponentially, and the solution approaches the corresponding smooth shock wave as time tends to infinity.6.We give a mathematical definition of the entropy function and prove the equivalence of the existence of an entropy function and the symmetrization of the system. Next, we formulate the stability condition, and under that condition we show the global existence and asymptotic decay of smooth solutions in Sobolev spaces. Furthermore, we observe that the solution is time-asymptotically approximated by the solution to the eorresponding hyperbolic-parabolic coupled system. These results are applicable to the equation of a radiating gas.