The existence of Leray-Hopf weak solutions with linear strain

Abstract

This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in $\mathbb{R}^{n}$ ($n \in \mathbb{Z}$, $n\geq 2$). Concerning initial data of the form $Ax+v(0)$, where $A \in M_{n}(\mathbb{R})$ and $v(0) \in L^{2}_{\sigma}(\mathbb{R}^{n})$, the weak solutions are properly-defined with the aid of the alternativity of the trilinear from $(Ax\cdot\nabla)v\cdot\varphi$. Furthermore, we construct the Leray-Hopf weak solution which satisfies not only the Navier-Stokes equations but also the energy inequality via the Galerkin approximation. From the viewpoint of quadratic forms, the Gronwall-Bellman inequality admits the uniform boundedness of the approximate solution.