对当前数学流体力学挑战的贡献

English | PDF (True) | 2004 | 159 Pages | ISBN : 3764371048 | 11.8 MB

This volume consists of five research articles, each dedicated to a significant topic in the mathematical theory of the Navier-Stokes equations, for compressible and incompressible fluids, and to related questions. All results given here are new and represent a noticeable contribution to the subject. One of the most famous predictions of the Kolmogorov theory of turbulence is the so-called Kolmogorov-obukhov five-thirds law. As is known, this law is heuristic and, to date, there is no rigorous justification. The article of A. Biryuk deals with the Cauchy problem for a multi-dimensional Burgers equation with periodic boundary conditions. Estimates in suitable norms for the corresponding solutions are derived for "large" Reynolds numbers, and their relation with the Kolmogorov-Obukhov law are discussed. Similar estimates are also obtained for the Navier-Stokes equation. In the late sixties J. L. Lions introduced a "perturbation" of the Navier­ Stokes equations in which he added in the linear momentum equation the hyper­ dissipative term (-Ll),Bu, f3 ~ 5/4, where Ll is the Laplace operator. This term is referred to as an "artificial" viscosity. Even though it is not physically moti­ vated, artificial viscosity has proved a useful device in numerical simulations of the Navier-Stokes equations at high Reynolds numbers. The paper of of D. Chae and J. Lee investigates the global well-posedness of a modification of the Navier­ Stokes equation similar to that introduced by Lions, but where now the original dissipative term -Llu is replaced by (-Ll)O:u, 0 S Ct < 5/4.

English | PDF（True）| 2004 | 159页| ISBN:3764371048| 11.8 MB本卷由五篇研究文章组成，每篇文章都专注于可压缩和不可压缩流体的Navier-Stokes方程数学理论中的一个重要主题以及相关问题。这里给出的所有结果都是新的，代表了对该主题的显著贡献。Kolmogorov湍流理论最著名的预测之一是所谓的Kolmogorov-obukhov五三定律。众所周知，这条定律是启发式的，到目前为止，还没有严格的理由。A.Biryuk的文章讨论了具有周期性边界条件的多维Burgers方程的Cauchy问题。推导了“大”雷诺数对应解的适当范数估计，并讨论了它们与柯尔莫哥洛夫-奥布霍夫定律的关系。Navier-Stokes方程也得到了类似的估计。在60年代末，J.L.Lions引入了Navier-Stokes方程的“扰动”，他在线性动量方程中添加了超耗散项（-Ll）Bu，f3~5/4，其中Ll是拉普拉斯算子。这个术语被称为“人工”粘度。尽管它不是物理运动的，但人工粘度已被证明是高雷诺数下Navier-Stokes方程数值模拟中的一种有用装置。D.Chae和J.Lee的论文研究了类似于Lions引入的Navier-Stokes方程的修正的全局适定性，但现在原始耗散项-Llu被（-Ll）O:u替换，0S Ct<5/4。

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