We consider {\it local} balances of momentum and angular momentum for the incompressible Navier-Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier-Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.

翻译：我们考虑不可压Navier-Stokes方程的动量与角动量局部平衡。首先，我们构建了这些物理量平衡（守恒律）的新弱形式，并证明其与常规守恒律表述等价。随后，我们证明采用EMAC非线性形式的Navier-Stokes方程连续Galerkin离散格式，无论是在欧拉框架还是拉格朗日框架下（离散后二者不再等价），都能保持弱形式守恒律的离散类似物。数值实验验证了这一新理论。