We consider a hyperbolic quasilinear version of the Navier–Stokes equations in \({\mathbb {R}}^2\), arising from using a Cattaneo type law instead of a Fourier law. These equations, which depend on a parameter \(\varepsilon \), are a way to avoid the infinite speed of propagation which occurs in the classical Navier–Stokes equations. We first prove the existence and uniqueness of solutions to these equations, and then exhibit smallness assumptions on the data, under which the solutions are global in time. In particular, these smallness assumptions disappear when \(\varepsilon \) vanishes, accordingly to the fact that the solutions of the 2D Navier–Stokes equations are global.

我们考虑\（{\ mathbb {R}} ^ 2 \）中Navier–Stokes方程的双曲拟线性版本，这是由于使用Cattaneo型定律而不是傅立叶定律引起的。这些方程取决于参数\（\ varepsilon \），是避免经典Navier–Stokes方程中发生无限传播速度的一种方法。我们首先证明这些方程解的存在性和唯一性，然后对数据展示较小的假设，在这些假设下，这些解决方案在时间上是全局的。尤其是，这些小假设在\（\ varepsilon \）消失时消失了，这与二维Navier–Stokes方程的解是全局的这一事实相对应。