Poincare--Riemann--Hilbert boundary-value problem for The Millennium Prize Problems

TL;DR

This paper explores the application of the Poincaré--Riemann--Hilbert boundary-value problem to complex issues like the Navier--Stokes equations and the Riemann zeta function, providing new methods for estimating solutions and zeros.

Contribution

It introduces a novel approach using the Poincaré--Riemann--Hilbert boundary-value problem to analyze the Navier--Stokes problem and the zeta function, offering effective solution estimates.

Findings

Effective estimates for Navier--Stokes solutions

Analysis of the zeros of the zeta function

Unified scheme for boundary-value problems

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. 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.