  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.
