On the Navier-Stokes problem

Alexander G Ramm

doi:10.24297/jam.v16i0.8088

Abstract

One of the millennium problems is discussed. The results of the author’s solution to this problem are explained. The problem discussed is the Navier-Stokes problem in the whole space.

Highlights

There is a large literature on the Navier-Stokes (NS) problem in R3 The global existence and uniqueness of a solution in R3 was not proved for a long time
The NS problem in R3 consists of solving the equations v + (v, ∇)v = −∇p + ν∆v + f, x ∈ R3, t ≥ 0, ∇ · v = 0, v(x, 0) = v0(x). (1)
The author’s results in [9] can be described as follows: a) The solution to NS problem exists for all t ≥ 0 in a suitable Banach space X of smooth functions and is unique in X. b) The solution depends continuously in a suitable norm on the data

Summary

Introduction

There is a large literature on the Navier-Stokes (NS) problem in R3 ( see [2], [3] and references therein.) The global existence and uniqueness of a solution in R3 was not proved for a long time. The new idea in the author’s solution to to NS problem is to use the Fourier transformed equation (2) and a new a priori estimate for v. The author’s results in [9] can be described as follows: a) The solution to NS problem exists for all t ≥ 0 in a suitable Banach space X of smooth functions and is unique in X. b) The solution depends continuously in a suitable norm on the data.

Objectives

Results

Conclusion