Reduction of modern problems of mathematics to the classical Riemann-Poincare-Hilbert problem

Authors

  • Asset Durmagambetov Institute of Information and Computational Technologies, International Science Complex ``Astana'', Kazakhstan

DOI:

https://doi.org/10.29020/nybg.ejpam.v11i4.3328

Keywords:

Schr¨odinger’s equation, potential, scattering amplitude, Cauchy problem, Navier--Stokes equations, Millennium Prize problems, Dirichlet, Riemann, Hilbert, Poincar´e Riemann hypothesis, zeta function, Hadamard

Abstract

Using the example of a complicated problem such as the Cauchy problem for the Navier--Stokes equation, we show how the Poincar\'e--Riemann--Hilbert boundary-value problem enables us to construct effective estimates of solutions for this case. The apparatus of the three-dimensional inverse problem of quantum scattering theory is developed for this. It is shown that the unitary scattering operator can be studied as a solution of the Poincar\'e--Riemann--Hilbert boundary-value problem. This allows us to go on to study the potential in the Schr\"odinger equation, which we consider as a velocity component in the Navier--Stokes equation. The same scheme of reduction of Riemann integral equations for the zeta function to the Poincar\'e--Riemann--Hilbert boundary-value problem allows us to construct effective estimates that describe the behaviour of the zeros of the zeta function very well.

Author Biography

  • Asset Durmagambetov, Institute of Information and Computational Technologies, International Science Complex ``Astana'', Kazakhstan
    Institute of Information and Computational Technologies, International Science Complex ``Astana'', Kazakhstan

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Published

2018-11-15

Issue

Section

Complex Analysis

How to Cite

Reduction of modern problems of mathematics to the classical Riemann-Poincare-Hilbert problem. (2018). European Journal of Pure and Applied Mathematics, 11(4), 1143-1176. https://doi.org/10.29020/nybg.ejpam.v11i4.3328