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

TL;DR

This paper introduces a novel approach using the Poincaré-Riemann-Hilbert boundary value problem to derive effective estimates for the solutions of the Navier-Stokes equations, linking quantum scattering theory and zeta-function analysis.

Contribution

It develops a new method to estimate Navier-Stokes solutions by connecting boundary value problems with quantum scattering and zeta-function analysis.

Findings

Effective estimates for Navier-Stokes solutions are constructed.

The approach links quantum scattering theory to fluid dynamics.

Zeros of the zeta function are described using this framework.

Abstract

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.