Reduction of modern mathematical problems to the classical Riemann--Poincaré--Hilbert problem.

Using the example of a complicated problem such as the Cauchy problem for the Navier--Stokes equation, we show how the Poincaré--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 Poincaré--Riemann--Hilbert boundary-value problem. This allows us to go on to study the potential in the Schrödinger 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 Poincaré--Riemann--Hilbert boundary-value problem allows us to construct effective estimates that describe the behaviour of the zeros of the zeta function very well. [ABSTRACT FROM AUTHOR]