Abstract:
Accumulated facts and information about the Navier-Stokes equations, together with a large number of
experiments and approximate calculations, made it possible to reveal some discrepancies between the
mathematical model of a viscous melt and real phenomena in the nature of real molten systems. There
are many reasons for this. One of them is the nonlinearity of the Navier-Stokes equations. And for nonlinear
equations it is known that in non-stationary problems, a solution satisfying it can exist not on
the entire interval t 0. Over a finite period of time, it can either go to infinity, or crumble.A solution
lose regularity and no satisfying the equations and begin branching. It is mathematically proved that if
this solution exists for t 0, then it may not seek to solve the stationary problem when stabilizing the
boundary conditions and external influences. The solutions of the nonstationary problem obtained even
with a smooth initial regime and smooth external influences can become less regular with time, and then
generally go into irregular or turbulent regimes. The actual implementation of this or that branch of
the solution depends on extraneous reasons not taken into account in the Navier-Stokes equations. In the
proposed paper, we constructed a numerical scheme with good convergence. The regularization of the initial
systems of differential equations by "-approximation is constructed. The Galerkin method is implemented
ensuring the correctness of boundary value problems for an incompressible viscous flow both numerically and
analytically. A splitting scheme for the Navier-Stokes equations with a weak approximation is constructed.
An approximation is constructed for stationary and nonstationary models of an incompressible melt, which
leads to nonlinear equations of hydrodynamics to a system of equations of Cauchy-Kovalevskaya type.
