New global estimations of the Cauchy problem for the Navier-Stokes equations

Using the example of such a complicated problem 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 developing for this. In which 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 Schrodinger 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 behavior of the zeros of the zeta function very well.