TY - JOUR

T1 - Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics

AU - Giesselmann, Jan

AU - Lattanzio, Corrado

AU - Tzavaras, Athanasios

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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.

PY - 2016/11/18

Y1 - 2016/11/18

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/583110

UR - http://arxiv.org/abs/1510.00801

UR - http://www.scopus.com/inward/record.url?scp=84995752705&partnerID=8YFLogxK

U2 - 10.1007/s00205-016-1063-2

DO - 10.1007/s00205-016-1063-2

M3 - Article

VL - 223

SP - 1427

EP - 1484

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 1432-0673

IS - 3

ER -