On the Caginalp phase-field system based on the Cattaneo law with nonlinear coupling

Ovono, Armel Andami; Miranville, Alain

doi:10.3934/math.2016.1.24

Cited by 3 publications

(1 citation statement)

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“…We refer, for instance, to [36,28,5] for the approximation of the incompressible Navier-Stokes equations by using Oldroyd-type constitutive laws with relaxation, to [10,35] for the approximation of the Timoshenko-Fourier system by the Timoshenko-Cattaneo system, and to [13,14] for the local convergence of hyperbolic-parabolic systems to the full compressible Navier-Stokes equations. See also [3,30,27,34] on this topic. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Relaxed Euler systems and convergence to Navier-Stokes equations

Peng¹

2021

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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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.

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