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Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations
[Existence et stabilité de solutions non caractéristiques et stationnaires des équations de Navier–Stokes non isentropiques sur un intervalle]
Blake Barker1 ; Benjamin Melinand2 ; Kevin Zumbrun3
1 Brigham Young University, Provo, UT 84602. Research partially supported under NSF grant no. DMS-140087
2 CEREMADE, CNRS, Université Paris-Dauphine, Université PSL, 75016 PARIS, FRANCE
3 Indiana University, Bloomington, IN 47405. Research of K.Z. was partially supported under NSF grant no. DMS-1700279.
Confluentes Mathematici, Tome 15 (2023), pp. 1-26.

Nous étudions le problème du tube de choc, établissant l’existence et la stabilité de solutions stationnaires des équations de Navier–Stokes non isentropiques pour des données non caractéristiques. Nous présentons aussi des simulations numériques indiquant l’unicité et la stabilité de telles solutions. Dans le même temps, nous donnons un exemple d’équation d’état artificielle possédant une entropie convexe où l’unicité n’a pas lieu. Ce phénomène est associé à une instabilité et une bifurcation de Hopf de solutions périodiques en temps.

We treat the 1D shock tube problem, establishing existence of steady solutions of full (nonisentropic) polytropic gas dynamics with arbitrary noncharacteristic data. We present also numerical experiments indicating uniqueness and time-asymptotic stability of such solutions. At the same time, we give an example of an (artificial) equation of state possessing a convex entropy for which there holds nonuniqueness of solutions. This is associated with instability and Hopf bifurcation to time-periodic solutions.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/cml.90
Mots clés : Steady solutions, gas dynamics, Evans function
Blake Barker 1 ; Benjamin Melinand 2 ; Kevin Zumbrun 3

1 Brigham Young University, Provo, UT 84602. Research partially supported under NSF grant no. DMS-140087
2 CEREMADE, CNRS, Université Paris-Dauphine, Université PSL, 75016 PARIS, FRANCE
3 Indiana University, Bloomington, IN 47405. Research of K.Z. was partially supported under NSF grant no. DMS-1700279.
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Blake Barker; Benjamin Melinand; Kevin Zumbrun. Existence and stability of steady noncharacteristic solutions on a finite interval of full compressible Navier–Stokes equations. Confluentes Mathematici, Tome 15 (2023), pp. 1-26. doi : 10.5802/cml.90. https://cml.centre-mersenne.org/articles/10.5802/cml.90/

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