The existence of Leray-Hopf weak solutions with linear strain

This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in R (n ∈ Z, n ≥ 2). Concerning initial data of the form Ax + v(0), where A ∈ Mn(R) and v(0) ∈ Lσ(R), the weak solutions are properly-defined with the aid of the alternativity of the trilinear from (Ax ·∇)v ·φ. 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.