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

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How to Cite

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