On Masuda uniqueness theorem for Leray-Hopf weak solutions in mixed-norm spaces
The authors prove an uniqueness result for Leray-Hopf weak solutions to the Navier-Stokes equations \(u_{t}-\Delta u+(u\cdot \nabla )u+\nabla P=0\), \(\operatorname{div}(u)=0\), in \(\mathbb{R}^{d}\times (0,T)\), \(d\geq 3\), with the initial condition \(u(0,\cdot )=a_{0}(\cdot )\), where \(a_{0}\) is divergence-free. The main result of the paper proves that if \(\overrightarrow{p} =(p_{1},p_{2},\ldots ,p_{d})\in \lbrack 2,\infty )^{d}\) satisfies \(\frac{1}{ p_{1}}+\frac{1}{p_{2}}+\dots +\frac{1}{p_{d}}=1\) and \(p_{d}\neq 2\), and if \( u,v\) are two Leray-Hopf weak solutions of the above Navier-Stokes equations in \(\mathbb{R}^{d}\times (0,T)\), with \(u\in L_{\mathrm{loc}}^{\infty }((0,T),L^{ \overrightarrow{p}}(\mathbb{R}^{d}))\) and \(T\in (0,\infty ]\), starting from the same initial data \(a_{0}\in L_{\sigma }^{2}(\mathbb{R}^{d})\cap L_{\sigma }^{\overrightarrow{p}}(\mathbb{R}^{d})\), then \(u=v\) in \(\mathbb{R} ^{d}\times (0,T)\). Here \(L_{\sigma }^{2}(\mathbb{R}^{d})\) is the subspace of divergence-free functions in \(L_{\sigma }^{2}(\mathbb{R}^{d})\) and \[ L_{\sigma }^{\overrightarrow{p}}(\mathbb{R}^{d})=\{u\in L^{\overrightarrow{p} }(\mathbb{R}^{d})^{d}:\int_{\mathbb{R}^{d}}u(x)\cdot \nabla \varphi (x)dx=0, \quad \forall \varphi \in C_{0}^{\infty }(\mathbb{R}^{d})\}.\] The authors thus extend Masuda's uniqueness result obtained in the case \(d=3\) and \( p_{1}=p_{2}=\ldots =p_{d}\). They first recall the notion of Leray-Hopf weak solution as a function \(u\in L^{\infty }((0,T),L_{\sigma }^{2}(\mathbb{R} ^{d}))\) such that \(\nabla u\in L^{2}(\mathbb{R}^{d}\times (0,T))\) which satisfies a variational formulation associated to the above Navier-Stokes problem, has a bounded energy and satisfies regularity properties. The key tools for the proof are a Sobolev embedding theorem and estimates on the Stokes semigroup in the mixed-norm spaces \(L^{\overrightarrow{p}}(\mathbb{R} ^{d})\).
scientific article; zbMATH DE number 7440248
scientific article