A Simple A-Priory Estimate for 3D Stationary Navier-Stokes System Via Interpolation

TL;DR

This paper demonstrates how interpolation inequalities can simplify a priori estimates for solutions to the stationary Navier-Stokes system, especially in nonlinear problems involving bootstrap arguments.

Contribution

It introduces a straightforward interpolation approach to derive a priori estimates for 3D stationary Navier-Stokes solutions, simplifying traditional bootstrap methods.

Findings

Derived a priori estimate in smooth function spaces for stationary Navier-Stokes solutions.

Showed the applicability of interpolation inequalities to nonlinear PDE estimates.

Potentially simplified the analysis of regularity for fluid dynamics problems.

Abstract

This short communication is motivated by a paper by O.A.Ladyzhenskaya, where a simple interpolation inequality was proved between summable smooth spaces. Such interpolation was applied as a technical tool for obtaining estimates of the solution to the linear Stokes system. But such cooperative simultaneous applying summable and smooth functional spaces even more often occurs at investigations of nonlinear problems, where, in particular, some bootstrap arguments are often involved. And we believe that the taking into account such interpolation reasoning can notably simplify different bootstrap procedures to rise up the smoothness of a solution or obtain an a-priory estimate in smooth classes of functions. Therefore by the present paper we just would like to attract the attention of the reader to such possibility. And for this we are going to demonstrate an obtaining of the a-priory estimate in a smooth space to the solution of the stationary Navier-Stokes system in a bounded (for simplicity) domain.