Hyperbolic Quasilinear Navier–Stokes Equations in ${\mathbb {R}}^2$

Abstract

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.