Nonlinear Generalized Functions: their origin, some developments and recent advances
Nonlinear Generalized Functions: their origin, some developments and recent advances
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非线性广义函数：它们的起源、一些发展和最新进展
DOI：
10.11606/issn.2316-9028.v7i2p201-239
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发表时间：
2014
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通讯作者：
J. Colombeau
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作者：
J. Colombeau
我们揭示了数学与真实的世界之间相互作用的一些简单事实，证明了数学对象“非线性广义函数”，它们是模拟真实的世界所必需的，迄今为止似乎被数学家们普遍忽视了。然后，我们描述了如何“非线性理论的广义函数”内获得的Leopoldo Nachbin组的无限维全纯1980年和1985年之间。这一新的理论允许任意分布相乘，并包含了上述数学对象，这表明这一理论的特征对于真实的世界的数学描述是自然的和不可避免的。最后，我们提出了直接应用的理论，如存在唯一性的系统的偏微分方程没有经典的解决方案和计算的冲击波的非发散形式的系统在1985年和1995年之间完成的，我们给出了三个不同性质的例子：弹性，宇宙学，多流体流动。
We expose some simple facts at the interplay between mathematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally neglected up to now by mathematicians. Then we describe how a "nonlinear theory of generalized functions" was obtained inside the Leopoldo Nachbin group of infinite dimensional holomorphy between 1980 and 1985. This new theory permits to multiply arbitrary distributions and contains the above mathematical objects, which shows that the features of this theory are natural and unavoidable for a mathematical description of the real world. Finally we present direct applications of the theory such as existence-uniqueness for systems of PDEs without classical solutions and calculations of shock waves for systems in non-divergence form done between 1985 and 1995, for which we give three examples of different nature: elasticity, cosmology, multifluid flows.
坎皮纳斯大学（巴西）
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