Contributions to Current Challenges in Mathematical Fluid Mechanics (eBook, PDF)

Redaktion: Galdi, Giovanni P.; Rannacher, Rolf; Heywood, John G.

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Birkhäuser Basel
• Seitenzahl: 152
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9783034878777
• Artikelnr.: 53160268

Autorenporträt

The mathematical theory of the Navier-Stokes equations presents fundamental open questions that represent many challenges for the interested mathematicians. This volume contains a series of articles whose objective is to furnish new contributions and ideas to these questions, with particular regard to turbulence modelling, regularity of solutions to the initial-value problem, flow in region with an unbounded boundary, and compressible flow.

Inhaltsangabe

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.