The book provides a comprehensive analysis of equations, catering to both beginners and advanced readers. It covers a wide range of topics, presenting multiple perspectives and extensive bibliographic resources for further exploration. This thorough approach ensures that readers can deepen their understanding of the subject matter effectively.
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.
This book surveys a broad range of research results on the physical and mathematical modeling, as well as the numerical simulation of hemodynamical flows, that is, of the complex fluid and structural mechanical processes that occur in the human blood circulation system. The topics treated include continuum mechanical description; choice of suitable liquid and wall models; mathematical analysis of coupled models; numerical methods for flow simulation; parameter identification and model calibration; fluid-solid interaction; mathematical analysis of piping systems; mesh and mode adaptivity; particle transport in channels and pipes; artificial boundary conditions, and many more. Hemodynamics is an area of active current research, and this book provides an entry into the field for graduate students and researchers. Hemodynamical Flows was developed out of a series of lectures presented by the authors at the Oberwolfach Research Institute (MFO) in Oberwolfach-Walke, Germany, in November, 2005.
This volume consists of six articles, each treating an important topic in the theory ofthe Navier-Stokes equations, at the research level. Some of the articles are mainly expository, putting together, in a unified setting, the results of recent research papers and conference lectures. Several other articles are devoted mainly to new results, but present them within a wider context and with a fuller exposition than is usual for journals. The plan to publish these articles as a book began with the lecture notes for the short courses of G. P. Galdi and R. Rannacher, given at the beginning of the International Workshop on Theoretical and Numerical Fluid Dynamics, held in Vancouver, Canada, July 27 to August 2, 1996. A renewed energy for this project came with the founding of the Journal of Mathematical Fluid Mechanics, by G. P. Galdi, J. Heywood, and R. Rannacher, in 1998. At that time it was decided that this volume should be published in association with the journal, and expanded to include articles by J. Heywood and W. Nagata, J. Heywood and M. Padula, and P. Gervasio, A. Quarteroni and F. Saleri. The original lecture notes were also revised and updated.