Fast Singular Limits of Hyperbolic PDEs
Fast Singular Limits of Hyperbolic PDEs
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DOI：
10.1006/jdeq.1994.1157
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发表时间：
1994-12
影响因子：
2.4
通讯作者：
S. Schochet
中科院分区：
文献类型：
--
作者：
S. Schochet
微分方程中存在小参数通常表明其解取决于两个不同的时间尺度。当我们对较长时间尺度上解的行为感兴趣时，参数趋于零的极限称为奇异，因为短尺度上的振荡可能会阻止解的收敛。如果解确实收敛，即使在时间上不均匀，那么奇异极限将被称为“慢”，而如果领先的小参数渐近仍然依赖于短时间尺度，那么极限就是“快”。慢奇异极限包括像 ɛ (u,–u,,)--u,--cu,= 0 这样的极限，其中短尺度是耗散的，因此仅在初始层中很重要。涉及小扰动的长期影响的问题（如 u,–u.,,= cuj 的 t= O (1/e) 行为）很快，因为 O (1) 时间尺度上的振荡持续存在。对于具有大常数项的拟线性对称双曲系统，
The presence of a small parameter in a differential equation often indicates that its solutions depend on two different time scales. When we are interested in the behavior of solutions over the longer time scale then the limit as the parameter tends to zero is called singular, because oscillations on the short scałe might prevent convergence of solutions. If solutions nonetheless do converge, even nonuniformly in time, then the singular limit will be termed “slow,” while if the leading small-parameter asymptotics retain dependence on the short time scale then the limit is “fast.” Slow singular limits include those like ɛ (u,–u,,)--u,--cu,= 0 in which the short scale is dissipative, and so is important only in an initial layer. Problems involving the long-term influence of small perturbations, as in the t= O (1/e) behavior of u,–u.,,= cuj, are fast because oscillations on the O (1) time scale persist. For quasilinear symmetric-hyperbolic systems with large constant-coefficient terms,