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arXiv 1909.08271gr-qc
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关于相对论性热方程的不适定性和稳定性

On the illposedness and stability of the relativistic heat equation

A. L. García-Perciante, Oscar. Reula

摘要:

本文通过一个简单的例子分析了抛物演化与广义协方差的不相容性。为此,我们引入了一个单位时间类四向量,并研究了关于它的最简单的热通量方程.在这种情况下,该向量场是表面形成然后局部高波数极限显示适定性,但一旦失去该属性的柯西问题成为不适定.我们还讨论了如何Maxwell-Cattaneo型修改的系统,使它很好地提出和链接的振幅的修改,这是有关的所谓的第二波的速度的系统,表面正交性的失败的大小。

英文摘要:

In this note we analyze, in terms of a simple example, the incompatibility of parabolic evolution and general covariance. For this we introduce a unit time-like four-vector and study the simplest heat flux equation with respect to it. In cases where this vector field is surface forming then the local high wave number limit shows well posedness, but as soon as that property is lost the Cauchy problem becomes ill-posed. We also discuss how the Maxwell-Cattaneo type modification of the system renders it well posed and link the amplitude of the modification, which is related to the so-called second wave speed of the system, to the size of the failure of surface orthogonality.

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