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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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DOI : 10.1016/j.anihpc.2020.07.007
Keywords: Compressible and incompressible Navier-Stokes equations, Newtonian fluid, Relaxed Euler systems, Local and global convergence
Peng, Yue-Jun 1
@article{AIHPC_2021__38_2_369_0,
author = {Peng, Yue-Jun},
title = {Relaxed {Euler} systems and convergence to {Navier-Stokes} equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {369--401},
publisher = {Elsevier},
volume = {38},
number = {2},
year = {2021},
doi = {10.1016/j.anihpc.2020.07.007},
mrnumber = {4211990},
zbl = {1458.35319},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.007/}
}
TY - JOUR AU - Peng, Yue-Jun TI - Relaxed Euler systems and convergence to Navier-Stokes equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2021 SP - 369 EP - 401 VL - 38 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.007/ DO - 10.1016/j.anihpc.2020.07.007 LA - en ID - AIHPC_2021__38_2_369_0 ER -
%0 Journal Article %A Peng, Yue-Jun %T Relaxed Euler systems and convergence to Navier-Stokes equations %J Annales de l'I.H.P. Analyse non linéaire %D 2021 %P 369-401 %V 38 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2020.07.007/ %R 10.1016/j.anihpc.2020.07.007 %G en %F AIHPC_2021__38_2_369_0
Peng, Yue-Jun. Relaxed Euler systems and convergence to Navier-Stokes equations. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 369-401. doi: 10.1016/j.anihpc.2020.07.007
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