The millennium problem of the Navier-Stokes equations
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 weak solutions to the Navier-Stokes equations.
scientific article; zbMATH DE number 7219509