Cauchy problem for a model system of the radiating gas: Weak solutions with a jump and classical solutions
Cauchy problem for a model system of the radiating gas: Weak solutions with a jump and classical solutions
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DOI：
10.1142/s0218202599000063
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发表时间：
1999-02
影响因子：
3.5
通讯作者：
S. Kawashima;S. Nishibata
中科院分区：
文献类型：
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作者：
S. Kawashima;S. Nishibata
本文讨论由一维辐射气体模型近似得到的系统初值问题解的整体存在性和时间渐近状态。当初始数据的空间导数大于某个负临界值时，在时间上全局存在唯一解。但如果它小于另一个负临界值，则相应解的空间导数在有限时间内爆破。因此，在适当的意义下考虑弱解是很自然的。作为弱解的原型，我们考虑了具有Riemann初值的Cauchy问题，其左状态大于右状态。这个条件保证了相应的行波的存在，渐近地连接左状态和右状态。当初始间断的大小小于1/2时，该Riemann问题存在整体弱解。虽然解具有间断性，但通过引入熵条件，我们得到了弱意义下解的唯一性。此外，当时间t趋于无穷大时，解中包含的不连续性的大小以指数速率衰减到零。同时，该解以t-1/4的速率一致地逼近相应的行波。第一个结果是由极大值原理得到的。为了证明第二个结果，我们使用了一种能量方法和一些估计，这些估计也是通过极大值原理得到的。
This paper deals with the global existence and the time asymptotic state of solutions to the initial value problems for the system derived from approximating a one-dimensional model of a radiating gas. When the spatial derivative of the initial data is larger than a certain negative critical value, a unique solution exists globally in time. But if it is smaller than another negative critical value, the spatial derivative of the corresponding solution blows up in a finite time. Thus it is natural to think about weak solutions in a suitable sense. As a prototype of weak solutions, we consider the Cauchy problem with the Riemann initial data of which the left state is larger than the right state. This condition ensures the existence of the corresponding traveling wave, connecting the left state and the right state asymptotically. This Riemann problem admits a global weak solution, provided that the magnitude of the initial discontinuity is smaller than 1/2. Although the solution has a discontinuity, we have the uniqueness of a solution in weak sense by imposing the entropy condition. Furthermore, the magnitude of the discontinuity contained in the solution decays to zero with an exponential rate as the time t goes to infinity. Also, the solution approaches the corresponding traveling wave with the rate t-1/4 uniformly. The first result is obtained by the maximal principles. To show the second result, we have used an energy method with some estimates, which are also obtained through maximal principles.