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Contributions to Current Challenges in Mathematical Fluid Mechanics (Hardcover, 2004 ed.)
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Contributions to Current Challenges in Mathematical Fluid Mechanics (Hardcover, 2004 ed.)
Series: Advances in Mathematical Fluid Mechanics
Expected to ship within 10 - 15 working days
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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 < 5/4.
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