The existence of Leray-Hopf weak solutions with linear strain (Q1797237)

The Navier-Stokes problem in \(\mathbb R^n\) is considered, with linear (in space) initial data, involving a \(n\times n\) real matrix \(A\), which verifies a sort of ``coercivity'' property -- see the relation (3.10). A linear substitution is used to get the initial value problem (1.2), where \(A\) appears in a momentum equation. \textit{M. Campiti} et al. [Commun. Pure Appl. Anal. 13, No. 4, 1613--1627 (2014; Zbl 1292.76028)] obtained the global existence and uniqueness of strong solutions to (1.2) when \(n=2\). In the present paper, the global existence of Leray-Hopf weak solutions to (1.2), verifying the energy inequality (1.4), is given for an arbitrary \(n\), by using the Gallerkin approximation and the Artzela-Ascloli theorem (with Cantor's diagonal argument). For this, the author used some results obtained in [\textit{K. Masuda}, Tohoku Math. J. (2) 36, 623--646 (1984; Zbl 0568.35077)], related with approximation of \(L^2\) and \(H^1\) functions by sequences of continuous functions. The Friedrichs inequality for the corresponding triliniar form appearing in (1.2) is used. The energy inequality is obtained by using (also) the Bessel inequality. All proofs are very elegant and this paper can be considered as a step ahead for understanding the behavior of the Navier-Stokes solutions.