Comments on the Navier–Stokes Problem

Alexander G Ramm

doi:10.3390/axioms10020095

Abstract

The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)≢0, ∇·v(x,0)=0 and the solution to the NSP exists for all t≥0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t≥0 and is equal to zero, v(x,t)≡0. Thus, one of the millennium problems is solved.

Highlights

IntroductionThe results of this paper are proved in detail in the monograph [1]. In the author’s papers, listed in the References (see [2,3,4,5]), some preliminary results are obtained
(a) First we reduce the Navier–Stokes problem (NSP) to an equivalent integral equation
The NSP paradox impies the conclusions we have made: The NSP is physically not a correct description of motion of incompressible viscous fluid in R3 and the NSP does not have a solution on the interval [0, ∞) unless the data are equal to zero

Summary

Introduction

The results of this paper are proved in detail in the monograph [1]. In the author’s papers, listed in the References (see [2,3,4,5]), some preliminary results are obtained. (b) Secondly, we prove that any solution to Equation (7) satisfies integral inequality (15), see below. We prove that Equation (17) has a solution in the space C (R+ ), supt≥0 q(t) < c, provided that the data v0 ( x ) is smooth and rapidly decaying at infinity. This solution is unique and q(0) = 0. The NSP paradox impies the conclusions we have made: The NSP is physically not a correct description of motion of incompressible viscous fluid in R3 and the NSP does not have a solution on the interval [0, ∞) unless the data are equal to zero.

Derivation of the Integral Inequality

Uniqueness of the Solution to the NSP

Conclusions