Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics

Handle URI:
http://hdl.handle.net/10754/583110
Title:
Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics
Authors:
Giesselmann, Jan; Lattanzio, Corrado; Tzavaras, Athanasios ( 0000-0002-1896-2270 )
Abstract:
For an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws: we prove stability for the NSK system via a modified relative energy approach. We prove continuous dependence of solutions on initial data and convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, but compensating via higher-order gradients.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Springer Nature
Journal:
Archive for Rational Mechanics and Analysis
Issue Date:
26-Oct-2016
DOI:
10.1007/s00205-016-1063-2
ARXIV:
arXiv:1510.00801
Type:
Article
ISSN:
1432-0673; 0003-9527
Sponsors:
JG partially supported by the German Research Foundation (DFG) via SFB TRR 75 `Tropfendynamische Prozesse unter extremen Umgebungsbedingungen'. AET acknowledges the support of the King Abdullah University of Science and Technology (KAUST) and of the Aristeia program of the Greek Secretariat for Research through the project DIKICOMA.
Additional Links:
http://arxiv.org/abs/1510.00801
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorGiesselmann, Janen
dc.contributor.authorLattanzio, Corradoen
dc.contributor.authorTzavaras, Athanasiosen
dc.date.accessioned2016-11-13T07:34:48Z-
dc.date.available2015-12-02T12:44:51Zen
dc.date.available2016-11-13T07:34:48Z-
dc.date.issued2016-10-26-
dc.identifier.issn1432-0673en
dc.identifier.issn0003-9527en
dc.identifier.doi10.1007/s00205-016-1063-2en
dc.identifier.urihttp://hdl.handle.net/10754/583110-
dc.description.abstractFor an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws: we prove stability for the NSK system via a modified relative energy approach. We prove continuous dependence of solutions on initial data and convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, but compensating via higher-order gradients.en
dc.description.sponsorshipJG partially supported by the German Research Foundation (DFG) via SFB TRR 75 `Tropfendynamische Prozesse unter extremen Umgebungsbedingungen'. AET acknowledges the support of the King Abdullah University of Science and Technology (KAUST) and of the Aristeia program of the Greek Secretariat for Research through the project DIKICOMA.en
dc.language.isoenen
dc.publisherSpringer Natureen
dc.relation.urlhttp://arxiv.org/abs/1510.00801en
dc.rightsThe final publication is available at Springer via http://dx.doi.org/10.1007/s00205-016-1063-2en
dc.titleRelative energy for the Korteweg theory and related Hamiltonian flows in gas dynamicsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalArchive for Rational Mechanics and Analysisen
dc.eprint.versionPost-printen
dc.contributor.institutionInstitute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 D-70563 Stuttgart Germanyen
dc.contributor.institutionDipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universita degli Studi dell’Aquila Via Vetoio I-67010 Coppito (L’Aquila) AQ Italyen
dc.contributor.institutionInstitute for Applied and Computational Mathematics Foundation for Research and Technology GR 70013 Heraklion, Crete Greeceen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
dc.identifier.arxividarXiv:1510.00801-

Version History

VersionItem Editor Date Summary
2 10754/583110grenzdm2016-11-13 07:28:33.477Post-print file submitted by Prof. Tzavaras via email.
1 10754/583110.1wangh0e2015-12-02 12:44:51.0
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