The millennium problem of the Navier-Stokes equations
type: Item
instance of: scholarly article
title: The millennium problem of the Navier-Stokes equations
zbMATH Open document ID: 1448.35362
author: Reinhard Farwig
published in: Mitteilungen der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 2020-07-10T00:00:00Z
review text: The author presents a state-of-the-art review concerning the regularity of solutions to the Navier-Stokes equations. He introduces the space \(\mathcal{LH}_{T}=L^{\infty }(0,T;L_{\sigma }^{2}(\Omega ))\cap L^{2}(0,T;H_{0}^{1}(\Omega ))\), where \(L_{\sigma }^{2}(\Omega )\) is the subspace of \(L^{2}(\Omega )\) of divergence-free functions which satisfy \( v\cdot n=0\) on the boundary \(\partial \Omega \) of \(\Omega \). He presents the weak Leray-Hopf solutions as functions in \(\mathcal{LH}_{T}\) which satisfy the variational formulation \(-\int_{0}^{T}\int_{\Omega }u\cdot\varphi _{t}dxd\tau +\nu \int_{0}^{T}\int_{\Omega }\nabla u:\nabla \varphi dxd\tau -\int_{0}^{T}\int_{\Omega }u\otimes u:\nabla \varphi dxd\tau =\int_{\Omega }u_{0}\cdot \varphi (0,x)dx\), for every divergence-free and smooth function \(\varphi \) with compact support in \(\Omega \times \lbrack 0;T)\). For the existence of weak solutions, the author recalls that the critical point is linked to the regularity of the term \(u\cdot \nabla u\) which differs between the 2D and 3D cases. He introduces the Serrin number \( \mathcal{S}(v)=\frac{2}{s}+\frac{d}{q}\), associated to functions \(v\in L^{s}(0,T;L^{q}(\mathbb{R}^{d}))\), and the problems associated to its value with respect to 1 or 3/2. He presents the possibilities to build regular solutions for example through the Leray principle of construction and the approach based on the semigroup \(u(t)=e^{-\nu tA}u_{0}-\int_{0}^{T}e^{-\nu (t-\tau )A}P(u\cdot \nabla u)(\tau )d\tau \), which involves the Helmholtz projection operator \(P\) and the Stokes operator \(A\) and which may involve Besov spaces. He indicates the difficulty to use the Galerkin method or the semigroup approach to build strong solutions in \(C^{0}([0;T];L^{3}(\Omega))\subset L^{\infty }(0;T;L^{3}(\Omega ))\). He then presents results concerning the local regularity and singularities and he discusses an example. The paper ends with considerations on non-uniqueness results concerning w…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 35-02
zbMATH DE Number: 7219509
zbMATH Keywords: millennium problem
zbMATH Keywords: Navier-Stokes-equations
zbMATH Keywords: weak solution
zbMATH Keywords: regularity
zbMATH Keywords: Serrin number
zbMATH Keywords: Leray-Hopf solution
zbMATH Keywords: Helmholtz projection operator
zbMATH Keywords: Stokes operator
zbMATH Keywords: semigroup
MaRDI profile type: Publication
full work available at URL: dmvm-2020-0008
OpenAlex ID: W3040982911
DOI: 10.1515/DMVM-2020-0008
description: scientific article; zbMATH DE number 7219509