A nonlinear parabolic system in the theory of combustion
A nonlinear parabolic system in the theory of combustion
复制标题
燃烧理论中的非线性抛物线系统
DOI：
10.1090/qam/463631
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发表时间：
1975
期刊：
影响因子：
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通讯作者：
D. Sattinger
中科院分区：
文献类型：
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作者：
D. Sattinger
在本文中我们将假设 T0(x) > T0。边界初值问题(1.1)—(1.3)仅构成燃烧问题的最简单模型。关于化学动力学一般方程推导的讨论可以在 Gavalas 的书中找到 [7]。 Gel'fand [6] 和 [1] 的文章中讨论了特定模型 (1.1)(1.3)。正如我们在第二节中看到的。由图2可知，边界初值问题(1.1)—(1.3)的解T(x, t), n(x, t)渐近趋于唯一稳态T(x) = T0, n(x) = 0，且该最终状态与初始数据无关。这个问题的重点不在于系统最终达到的最终状态，而在于达到该状态的方式。事实上，系统的初始行为可能会显着不同，具体取决于初始数据。令 a = E/RT0 并输入 « = exp ( — a)。在实际问题中，a 的范围通常为 20 到 100 [8]，因此我们在本文中假设 e 是一个小参数（尽管不一定像 exp ( — 20) 那么小）。如果初始温度和浓度“低于临界点”，则燃烧在 t 量级的时间尺度上进行得非常缓慢，温度保持相对接近 T0，并且温度和浓度按 exp (—et) 衰减。另一方面，如果温度和
We shall assume throughout this paper that T0(x) > T0. The boundary-initial value problem (1.1)—(1.3) constitutes only the simplest model for combustion problems. A discussion of the derivation of the general equations of chemical kinetics may be found in the book by Gavalas [7]. The particular model (1.1)(1.3) has been discussed in an article by Gel'fand [6], and in [1]. As we see in Sec. 2, the solution T(x, t), n(x, t) of the boundary-initial value problem (1.1)—(1.3) tends asymptotically to the unique steady state T(x) = T0, n(x) = 0, and this ultimate state is independent of the initial data. The interest in the problem lies not in the final state which the system eventually reaches, but rather the manner in which that state is attained. In fact, the initial behavior of the system can differ markedly, depending on the initial data. Let a = E/RT0 and put « = exp ( — a). In actual problems a typically may range from 20 to 100 [8], and so we shall assume in this article that e is a small parameter (though not necessarily so small as exp ( — 20)). If the initial temperature and concentration are "below criticality", the combustion proceeds very slowly, on a time scale of order t, with the temperature remaining relatively close to T0, and the temperature and concentration decaying as exp (—et). On the other hand, if the temperature and