Contributions to Current Challenges in Mathematical Fluid Mechanics, Softcover reprint of the original 1st ed. 2004 Advances in Mathematical Fluid Mechanics Series

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.

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 Navier—Stokes system.- References.- On the Global Well-posedness and Stability of the Navier—Stokes and the Related Equations.- 1. Introduction.- 2. Littlewood—Paley decomposition.- 3. Proof of Theorems.- References.- The Commutation Error of the Space Averaged Navier—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))d—norm of the commutation error term.- 5. Error estimates in the (H-1(?))d—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 ñ in some LES models.- References.- The Nonstationary Stokes and Navier—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 Navier—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.

Covers a wide range of hot topics in the mathematical theory of the Navier-Stokes equations