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

#### 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.

#### 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