Navier-Stokes equations. On the existence and the search method for global solutions.

In this book, the variational principle of extremum for viscous incompressible and compressible fluid is formulated and proved. From this principle, it follows that the Navier-Stokes equations represent the extremum conditions of a certain functional. It is described the method of seeking solution for these equations, which consists in moving along the gradient to this functional extremum. The conditions of reaching this extremum are also formulated and they are at the same time the necessary and sufficient conditions of the existence of the global extremum of this functional. Next, some closed systems are considered. It is proved that for them the necessary and sufficient conditions of the global extremum for the aforementioned functional always exist. Accordingly, the search for the global extremum is always successful and so, the unique solution of the Navier-Stokes equations is actually found. It is contended that the systems, which are described by the Navier-Stokes equations with determined boundary conditions (pressure or speed) on all the boundaries, are the closed systems. It is also shown that such type of systems includes the following systems: (*) the systems bounded by impermeable walls, (**) the systems with free surfaces under action of some known pressure, (***) the systems with movable walls under action of some known pressure that are called generating surfaces, through which the fluid flow passes with some known speed. The book is supplemented by the corresponding program codes in the computer program MATLAB, namely by functions that realize the calculation method and the computer program test codes. References to the test program codes are given in the text of this book when the corresponding examples are described. All the computer programs in the MATLAB codes and their explanations are respectively attached in Appendices 8 and 9 of this book. The book is available in printed form (so-called print-on-demand). The author can send this book for $90. (including postage). Thank you for your interest! Sincerely, Solomon, solik@netvision.net.i