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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