The Navier-Stokes equations: existence theorems and energy equalities (Q656355)

This paper deals with a model system of generalized Navier-Stokes equations. Firstly, the authors investigate a special initial-boundary value problem in the parabolic cylinder \(Q_{T}= \Omega \times (0,T)\), where \(\Omega \subset {\mathbf R}^{d} (d \geq 2)\) denotes a bounded domain with Lipschitz boundary. Their concept of weak solutions involves two points: an integral identity and an energy inequality. The existence of weak solutions is proved by construction as the limit of solutions to regularized problems. Also, the convergence of fluxes and viscous energy densities can be proved for different dimensions \(d\). Finally, an associated stationary problem is studied.