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Download E-books Contributions to Current Challenges in Mathematical Fluid Mechanics (Advances in Mathematical Fluid Mechanics) PDF

This quantity includes 5 learn articles, each one devoted to an important subject within the mathematical idea of the Navier-Stokes equations, for compressible and incompressible fluids, and to similar questions. All effects given listed below are new and characterize a visible contribution to the topic. the most recognized predictions of the Kolmogorov idea of turbulence is the so-called Kolmogorov-obukhov five-thirds legislation. As is understood, this legislation is heuristic and, up to now, there isn't any rigorous justification. the thing of A. Biryuk offers with the Cauchy challenge for a multi-dimensional Burgers equation with periodic boundary stipulations. Estimates in appropriate norms for the corresponding ideas are derived for "large" Reynolds numbers, and their relation with the Kolmogorov-Obukhov legislation are mentioned. comparable estimates also are received for the Navier-Stokes equation. within the overdue sixties J. L. Lions brought a "perturbation" of the Navier­ Stokes equations within which he additional within the linear momentum equation the hyper­ dissipative time period (-Ll),Bu, f3 ~ 5/4, the place Ll is the Laplace operator. This time period is often called an "artificial" viscosity. although it isn't really bodily moti­ vated, man made viscosity has proved an invaluable machine in numerical simulations of the Navier-Stokes equations at excessive Reynolds numbers. The paper of of D. Chae and J. Lee investigates the worldwide well-posedness of a amendment of the Navier­ Stokes equation just like that brought through Lions, yet the place now the unique dissipative time period -Llu is changed through (-Ll)O:u, zero S Ct < 5/4.

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