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• Title: Application of Analytic Functions to the Global Solvabilty of the Cauchy Problem for Equations of Navier-Stokes (Report)
• Author : Asset Durmagambetov
• Release Date : January 01, 2010
• Genre: Engineering,Books,Professional & Technical,
• Pages : * pages
• Size : 86 KB

Description

1. INTRODUCTION The work of L. Fadeyev dedicated to the many- dimensional inverse problem of scattering theory inspired the author of this article to conduct this research. The first results obtained by the author are described in the works [1-3]. This problem includes a number of subproblems which appear to be very interesting and complicated. These subproblems are thoroughly considered in the works of the following scientists: R. Newton [4], R. Faddeyev [5], R. Novikov and G. Khenkin [6], A. Ramm [3] and others. The latest advances in the theory of SIPM (Scattering Inverse Problem Method) were a great stimulus for the author as well as other researchers. Another important stimulus was the work of M. Lavrentyev on the application of analytic functions to Hydrodynamics. Only one-dimensional equations were integrated by SIPM. The application of analytic functions to Hydrodynamics is restricted only by bidimensional problems. The further progress in applying SIPM to the solution of nonlinear equations in R3 was hampered by the poor development of the three-dimensional inverse problem of scattering in comparison with the progress achieved in the work on the one-dimensional inverse problem of scattering and also by the difficulties the researchers encountered building up the corresponding Lax' pairs. It is easy to come to a conclusion that all the success in developing the theory of SIPM is connected with analytic functions, i.e., solutions to Schrodinger's equation. Therefore we consider Schrodinger's equation as an interrelation between real-valued functions and analytic functions, where real-valued functions are potentials in Schrodinger's equation and analytic functions are the corresponding eigenfunctions of the continuous spectrum of Schrodinger's operator. The basic aim of the paper is to study this interrelation and its application for obtaining new estimates to the solutions of the problem for Navier-Stokes' equations. We concentrated on formulating the conditions of momentum and energy conservation laws in terms of potential instead of formulating them in terms of wave functions. As a result of our study, we obtained non-trivial nonlinear relationships of potential. The effectiveness and novelty of the obtained results are displayed when solving the notoriously difficult Chauchy problem for Navier-Stokes' equations of viscous incompressible fluid.

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