We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.

我们考虑了在可压缩和不可压缩情况下具有松弛的欧拉型系统对牛顿流体的Navier-Stokes方程的逼近。这需要将速度的二阶导数项分解为一阶项。通常的分解会导致带有张量变量的近似系统。我们使用Hurwitz-Radon矩阵构造带有向量变量的近似系统。这些系统以平衡定律的形式编写，并接受严格的凸熵，因此它们是对称的双曲型。对于平滑解，我们证明了均匀系统中近似系统对Navier-Stokes方程的收敛性。还显示了接近系统恒定平衡状态的初始数据的全局时间收敛。