Cover image for Contributions to current challenges in mathematical fluid mechanics
Contributions to current challenges in mathematical fluid mechanics
Advances in mathematical fluid me504 __ Includes bibliographical references.
Publication Information:
Basel : Birkhauser Verlag , 2004


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30000010082042 QA901 C66 2004 Open Access Book

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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

Table of Contents

On Multidimensional Burgers Type Equations with Small ViscosityA. Biryuk
On the Global Well-posedness and Stability of the Navier-Stokes and Related EquationsD. Chae and J. Lee
The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded DomainA. Dunca and V. John and W.J. Layton
The Nonstationary Stokes and Navier-Stokes Flows Through an ApertureT. Hishida
Asymptotic Behaviour at Infinity of Exterior Three-Dimensional Steady Compressible FlowT. Leonaviciene and K. Pileckas