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2004 | Book

On Multidimensional Burgers Type Equations with Small Viscosity Read chapter Read first chapter

Contributions to Current Challenges in Mathematical Fluid Mechanics

Editors: Giovanni P. Galdi, John G. Heywood, Rolf Rannacher

Publisher: Birkhäuser Basel

Book Series : Advances in Mathematical Fluid Mechanics

Included in: Professional Book Archive

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About this book

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.

Table of Contents

Frontmatter
On Multidimensional Burgers Type Equations with Small Viscosity
Abstract
We consider the Cauchy problem for a multidimensional Burgers type equation with periodic boundary conditions. We obtain upper and lower bounds for derivatives of solutions for this equation in terms of powers of the viscosity and discuss how these estimates relate to the Kolmogorov-Obukhov spectral law. Next we use the estimates obtained to get certain bounds for derivatives of solutions of the Navier-Stokes system.
Andrei Biryuk
On the Global Well-posedness and Stability of the Navier—Stokes and the Related Equations
Abstract
We study the problem of global well-posedness and stability in the scale invariant Besov spaces for the modified 3D Navier-Stokes equations with the dissipation term, −Δu replaced by \( {( - \Delta )^\alpha }u,0 \leqslant \alpha < \frac{5}{4} \). We prove the unique existence of a global-in-time solution in \( B_{2,1}^{\frac{5}{2} - 2\alpha } \) for initial data having small \( B_{2,1}^{\frac{5}{2} - 2\alpha } \) norm for all \( \alpha \in \left[ {0,\frac{5}{4}} \right) \). We also obtain the global stability of the solutions \( B_{2,1}^{\frac{5}{2} - 2\alpha } \) for \( \alpha \in \left[ {\frac{1}{2},\frac{5}{4}} \right) \). In the case \( \frac{1}{2} < \alpha < \frac{5}{4} \), we prove the unique existence of a global-in-time solution in \( B_{p,\infty }^{\frac{3}{p} + 1 - 2\alpha } \) for small initial data, extending the previous results for the case α = 1.
Dongho Chae, Jihoon Lee
The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain
Abstract
In Large Eddy Simulation of turbulent flows, the Navier-Stokes equations are convolved with a filter and differentiation and convolution are interchanged, introducing an extra commutation error term, which is nearly universally dropped from the resulting equations. We show that the commutation error is asymptotically negligible in L p (ℝ d )(i.e., it vanishes as the averaging radius S →0) if and only if the fluid and the boundary exert exactly zero force on each other. Next, we show that the commutation error tends to zero in H-1(Ω) as S →0. Convergence is proven also for a weak form of the commutation error. The order of convergence is studied in both cases. Last, we study the influence of the commutation error on the energy balance of the filtered equations.
A. Dunca, V. John, W. J. Layton
The Nonstationary Stokes and Navier-Stokes Flows Through an Aperture
Abstract
We consider the nonstationary Stokes and Navier—Stokes flows in aperture domains Ω ⊂ R n, n ≥ 3. We develop the L q -L r estimates of the Stokes semigroup and apply them to the Navier—Stokes initial value problem. As a result, we obtain the global existence of a unique strong solution, which satisfies the vanishing flux condition through the aperture and some sharp decay properties as t→∞, when the initial velocity is sufficiently small in the L n space. Such a global existence theorem is up to now well known in the cases of the whole and half spaces, bounded and exterior domains.
Toshiaki Hishida
Asymptotic Behavior at Infinity of Exterior Three-dimensional Steady Compressible Flow
Abstract
Steady compressible Navier-Stokes equations with zero velocity conditions at infinity are studied in a three-dimensional exterior domain. The case of small perturbations of large potential forces is considered. In order to solve the problem, a decomposition scheme is applied and the nonlinear problem is decomposed into three linear problems: Neumann-type problem, modified Stokes problem and transport equation. These linear problems are solved in weighted function spaces with detached asymptotics. The results on existence, uniqueness and asymptotics for the linearized problem and for the nonlinear problem are proved.
T. Leonavičienė, K. Pileckas
Metadata
Title
Contributions to Current Challenges in Mathematical Fluid Mechanics
Editors
Giovanni P. Galdi
John G. Heywood
Rolf Rannacher
Copyright Year
2004
Publisher
Birkhäuser Basel
Electronic ISBN
978-3-0348-7877-7
Print ISBN
978-3-0348-9606-1
DOI
https://doi.org/10.1007/978-3-0348-7877-7