Title	Contributions to Current Challenges in Mathematical Fluid Mechanics [electronic resource] / edited by Giovanni P. Galdi, John G. Heywood, Rolf Rannacher
Imprint	Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2004
Connect to	http://dx.doi.org/10.1007/978-3-0348-7877-7
Descript	VIII, 152 p. online resource

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 Navierยญ Stokes equations in which he added in the linear momentum equation the hyperยญ 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 motiยญ 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 Navierยญ 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

On Multidimensional Burgers Type Equations with Small Viscosity -- 1. Introduction -- 2. Upper estimates -- 3. Lower estimates -- 4. Fourier coefficients -- 5. Low bounds for spatial derivatives of solutions of the Navierโ{128}{148}Stokes system -- References -- On the Global Well-posedness and Stability of the Navierโ{128}{148}Stokes and the Related Equations -- 1. Introduction -- 2. Littlewoodโ{128}{148}Paley decomposition -- 3. Proof of Theorems -- References -- The Commutation Error of the Space Averaged Navierโ{128}{148}Stokes Equations on a Bounded Domain -- 1. Introduction -- 2. The space averaged Navier-Stokes equations in a bounded domain -- 3. The Gaussian filter -- 4. Error estimates in the (Lp(?d))dโ{128}{148}norm of the commutation error term -- 5. Error estimates in the (H-1(?))dโ{128}{148}norm of the commutation error term -- 6. Error estimates for a weak form of the commutation error term -- 7. The boundedness of the kinetic energy for รฑ in some LES models -- References -- The Nonstationary Stokes and Navierโ{128}{148}Stokes Flows Through an Aperture -- 1. Introduction -- 2. Results -- 3. The Stokes resolvent for the half space -- 4. The Stokes resolvent -- 5. L4-Lr estimates of the Stokes semigroup -- 6. The Navierโ{128}{148}Stokes flow -- References -- Asymptotic Behavior at Infinity of Exterior Three-dimensional Steady Compressible Flow -- 1. Introduction -- 2. Function spaces and auxiliary results -- 3. Stokes and modified Stokes problems in weighted spaces -- 4. Transport equation and Poisson-type equation -- 5. Linearized problem -- 6. Nonlinear problem -- References