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We consider {\it local} balances of momentum and angular momentum for theincompressible Navier-Stokes equations. First, we formulate new weak forms ofthe physical balances (conservation laws) of these quantities, and prove theyare equivalent to the usual conservation law formulations. We then show thatcontinuous Galerkin discretizations of the Navier-Stokes equations using theEMAC form of the nonlinearity preserve discrete analogues of the weak formconservation laws, both in the Eulerian formulation and the Lagrangianformulation (which are not equivalent after discretizations). Numerical testsillustrate the new theory.

Local conservation laws of continuous Galerkin method for the incompressible Navier--Stokes equations in EMAC form