Weak/BV Stability and a-contraction in fluid dynamics
[Le principe BV/faible et la théorie de a-contraction pour la dynamique des fluides]
Alexis Vasseur1
1Department of Mathematics, The University of Texas at Austin, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA
Journées équations aux dérivées partielles (2025), Exposé no. 10, 13 p.

This paper introduces the a-contraction theory for the $L^2$ stability of discontinuous solutions to conservation laws and their inviscid limits. It can be viewed as an extension of the Weak–Strong stability principle, first introduced by Dafermos and DiPerna, to the stability of discontinuous solutions–e.g. bounded in $BV$–among general weak entropic solutions. For example, small BV solutions to barotropic Euler equations can be obtained as inviscid limits of solutions to the associated compressible Navier–Stokes equations. In the barotropic setting, this extends the celebrated work of Bianchini and Bressan to models with physical viscosity.

Cet article présente la théorie de $a$‑contraction pour la stabilité en norme $L^2$ des solutions discontinues des lois de conservation et leurs limites asymptotiques. Elle peut être vue comme une extension du principe de stabilité fort/faible, introduit par Dafermos et DiPerna, à la stabilité des solutions discontinues–par exemple bornées en $BV$– au sein de la classe des solutions entropiques faibles générales. Par exemple, des solutions à petite variation totale ($BV$) des équations d’Euler barotropes peuvent être obtenues comme limites de viscosité évanescente de solutions des équations de Navier–Stokes compressibles associées. Dans le cadre barotrope, cela étend le célèbre travail de Bianchini et Bressan aux modèles avec viscosité physique.

Publié le :
DOI : 10.5802/jedp.701
Classification : 35L65, 76N10
Keywords: Stability, shock, a-contraction, Weak/BV principle, conservation Laws, fluid mechnics
Mots-clés : Stabilité, choc, a-contraction, principe BV/faible, Loi de conservation, méchanique des fluides

Alexis Vasseur  1

1 Department of Mathematics, The University of Texas at Austin, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA 
Alexis Vasseur. Weak/BV Stability and a-contraction in fluid dynamics. Journées équations aux dérivées partielles (2025), Exposé no. 10, 13 p.. doi: 10.5802/jedp.701
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