Relaxed Euler systems and convergence to Navier-Stokes equations

Abstract

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.

Global Convergence to Compressible Full Navier–Stokes Equations by Approximation with Oldroyd-Type Constitutive Laws

“…Meanwhile, the Navier-Stokes equations can also be approximated by first-order partial differential equations using different kinds of constitutive laws for non-Newtonian fluids. These approximate equations are referred to as relaxed Euler systems or hyperbolic Navier-Stokes equations, see for instance [33,34,16,30,10].…”

“…For the isentropic Navier-Stokes equations with constitutive law (1.7), the author of [42] obtained the local existence and the local convergence to the classical isentropic Navier-Stokes equations under condition tr(τ ) = 0, where tr(τ ) means the trace of matrix τ . In [30], the first author of the present paper constructed approximate systems with vector variables instead of tensor variables by using Hurwitz-Radon matrices in both compressible and incompressible cases. He proved the uniform (with respect to ε 1 and ε 2 ) global existence of smooth solutions near constant equilibrium state and the global-in-time convergence of the systems towards classical isentropic Navier-Stokes equations.…”