论文标题
在湍流理论中引起的某些抛物线系统的有限时间爆炸
Finite time blow-up for some parabolic systems arising in turbulence theory
论文作者
论文摘要
我们研究一类与湍流理论相关的非线性抛物线系统。这些系统可以看作是Prandtl单方程的简化版本和湍流的Kolmogorov双方程模型。 我们将注意力限制在一个空间维度的情况下。我们考虑扩散系数可能消失的初始数据。我们证明，在这种情况下，这些系统在足够高的规律性的Sobolev空间中局部良好，但也存在相应的解决方案在有限时间内爆炸的平滑初始数据。 我们能够证明两种不同类型的爆炸机制。此外，结果将扩展到运输扩散系统的情况，即考虑到对流的情况。
We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account.
