A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations

Didier Bresch; Frédéric Couderc; Pascal Noble; Jean-Paul Vila

doi:10.1016/j.crma.2015.09.020

Partial differential equations

A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations

Presented by: Olivier Pironneau

Didier Bresch ¹; Frédéric Couderc ²; Pascal Noble ²; Jean-Paul Vila ²

¹ LAMA – UMR5127 CNRS, bâtiment Le Chablais, campus scientifique, 73376 Le Bourget-du-Lac, France
² IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 39-43.

Abstract
Résumé

In this note, we propose a surprising and important generalization of the quantum Bohm potential identity. This formula allows us to design an original conservative extended formulation of Euler–Korteweg systems and the construction of a numerical scheme with entropy stability property under a hyperbolic CFL condition in the multi-dimensional setting. To the authors' knowledge, this generalization of the quantum Bohm identity strongly improves what is already known for simulation of such a dispersive system and is also important for theoretical studies on compressible Navier–Stokes equations with degenerate viscosities.

Dans cette note, on propose une importante généralisation de l'identité dite du potentiel de Bohm quantique. Cette dernière permet de définir une formulation augmentée des systèmes d'Euler–Korteweg, qui est sous forme conservative dans le cas multi-dimensionnel. Une conséquence très importante de cette formulation est la construction de schémas avec stabilité entropique sous condition CFL hyperbolique du système d'Euler–Korteweg. Cette généralisation de l'identité de Bohm évite donc le développement d'ondes parasites pour ces systèmes de type dispersif et est aussi importante, par exemple, dans l'étude des équations de Navier–Stokes compressibles à viscosités dégénérées.

Received: 2015-03-30
Accepted: 2015-09-24
Published online: 2015-11-10

DOI: 10.1016/j.crma.2015.09.020

Author's affiliations:

Didier Bresch ¹; Frédéric Couderc ²; Pascal Noble ²; Jean-Paul Vila ²

¹ LAMA – UMR5127 CNRS, bâtiment Le Chablais, campus scientifique, 73376 Le Bourget-du-Lac, France
² IMT, INSA Toulouse, 135, avenue de Rangueil, 31077 Toulouse cedex 9, France

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     author = {Didier Bresch and Fr\'ed\'eric Couderc and Pascal Noble and Jean-Paul Vila},
     title = {A generalization of the quantum {Bohm} identity: {Hyperbolic} {CFL} condition for {Euler{\textendash}Korteweg} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {39--43},
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     year = {2016},
     doi = {10.1016/j.crma.2015.09.020},
     language = {en},
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Didier Bresch; Frédéric Couderc; Pascal Noble; Jean-Paul Vila. A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 39-43. doi : 10.1016/j.crma.2015.09.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.020/

References
Cited by

[1] S. Benzoni-Gavage; R. Danchin; S. Descombes On the well-posedness for the Euler–Korteweg model in several space dimensions, Indiana Univ. Math. J., Volume 56 (2007) no. 4, pp. 1499-1579

[2] M. Boutounet; L. Chupin; P. Noble; J.-P. Vila Shallow water flows for arbitrary topography, Commun. Math. Sci., Volume 6 (2008) no. 1, pp. 73-90

[3] D. Bresch; B. Desjardins; C.K. Lin On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Differ. Equ., Volume 28 (2003) no. 3–4, pp. 843-868

[4] D. Bresch, A. Vasseur, C. Yu, A remark on the existence for degenerate compressible Navier–Stokes equations, 2015, in preparation.

[5] D. Bresch, F. Couderc, P. Noble, J.-P. Vila, Stable schemes for some compressible capillary fluid systems under hyperbolic Courant–Friedrichs–Lewy condition, in preparation.

[6] D. Bresch; B. Desjardins; E. Zatorska Two-velocity hydrodynamics in fluid mechanics: part II. Existence of global κ-entropy solutions to compressible Navier–Stokes systems with degenerate viscosities, J. Math. Pures Appl. (2015) (in press)

[7] M.A. Hoefer; M.J. Ablowitz; I. Coddington; E.A. Cornell; P. Engels; V. Schweikhard Dispersive and classical shock waves in Bose–Einstein condensates and gas dynamics, Phys. Rev. A, Volume 74 (2006) no. 2

[8] D. Jamet; D. Torres; J.U. Brackbill On the theory and computation of surface tension: the elimination of parasitic currents through energy conservation in the second-gradient method, J. Comput. Phys., Volume 182 (2002), pp. 262-276

[9] J. Liu; J.B. Schneider; J.P. Gollub Three-dimensional instabilities of film flows, Phys. Fluids, Volume 7 (1995) no. 1, pp. 55-67

[10] P. Noble; J.-P. Vila Stability theory for difference approximations of Euler–Korteweg equations and application to thin film flows, SIAM J. Numer. Anal., Volume 52 (2014) no. 6, pp. 2770-2791

[11] A. Vasseur; C. Yu Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, 2015 | arXiv

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