Skip to main content

Advertisement

SpringerLink
Log in

Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider a Euler system with dynamics generated by a potential energy functional. We propose a form for the relative energy that exploits the variational structure and we derive a relative energy identity. When applied to specific energies, this 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, and prove stability for the NSK system via a modified relative energy approach. We prove the continuous dependence of solutions on initial data and the 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, compensated by higher-order gradients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Antonelli P., Marcati P.: On the finite energy weak solutions to a system in quantum fluid dynamics. Commun. Math. Phys. 287, 657–686 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Antonelli P., Marcati P.: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203, 499–527 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benzoni-Gavage S., Danchin R., Descombes S., Jamet D.: Structure of Korteweg models and stability of diffuse interfaces. Interf. Free Bound. 7, 371–414 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benzoni-Gavage S., Danchin R., Descombes S.: On the well-posedness for the Euler–Korteweg model in several space dimensions. Indiana Univ. Math. J. 56, 1499–1579 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benzoni-Gavage, S.: Propagating phase boundaries and capillary fluids. Lecture Notes. 3ème cycle. Levico, 2010, pp. 57

  6. Brandon, D., Lin, T., Rogers, R.C.: Phase transitions and hysteresis in nonlocal and order-parameter models. Meccanica 30, 541–565 (1995). Microstructure and phase transitions in solids (Udine, 1994)

  7. Braack M., Prohl A.: Stable discretization of a diffuse interface model for liquid-vapor flows with surface tension. ESAIM: Math. Model. Numer. Anal. 47, 401–420 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ballew J., Trivisa K.: Weakly dissipative solutions and weak-strong uniqueness for the Navier–Stokes–Smoluchowski system. Nonlinear Anal. 91, 1–19 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brenier Y.: Topology-preserving diffusion of divergence-free vector fields and magnetic relaxation. Commun. Math. Phys. 330, 757–770 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Charve F.: Local in time results for local and non-local capillary Navier–Stokes systems with large data. J. Differ. Equ. 256, 2152–2193 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dafermos C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dafermos C.M.: Stability of motions of thermoelastic fluids. J. Thermal Stress. 2, 127–134 (1979)

    Article  Google Scholar 

  13. Dafermos C.M.: Quasilinear hyperbolic systems with involutions. Arch. Ration. Mech. Anal. 94, 373–389 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (2010)

  15. DiPerna R.J.: Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J. 28, 137–188 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donatelli D., Feireisl E., Marcati P.: Well/ill posedness for the Euler–Korteweg–Poisson system and related problems. Commun. Partial Differ. Equ. 40, 1314–1335 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry-Methods and Applications. Part II. The Geometry and Topology of Manifolds. Graduate Texts in Mathematics, Vol. 104. Springer, New York 1985

  18. Dunn J.E., Serrin J.: On the thermomechanics of interstitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Engel P., Viorel A., Rohde C.: A low-order approximation for viscous-capillary phase transition dynamics. Port. Math. 70, 319–344 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feireisl E., Novotný A.: Weak-strong uniqueness property for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gelfand, I.M., Fomin, S.V.: Calculus of Variations, Prentice Hall, New York 1963

  22. Giesselmann J.: A relative entropy approach to convergence of a low order approximation to a nonlinear elasticity model with viscosity and capillarity. SIAM J. Math. Anal. 46, 3518–3539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Giesselmann J., Makridakis C., Pryer T.: Energy consistent dg methods for the Navier–Stokes–Korteweg system. Math. Comput. 83, 2071–2099 (2014)

    Article  MATH  Google Scholar 

  24. Hamilton R.S.: The inverse function theorem of Nash and Moser. Bull. AMS 7, 65–222 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ju Q., Li H., Li Y., Jiang S.: Quasi-neutral limit of the two-fluid Euler–Poisson system. Commun. Pure Appl. Anal. 9, 1577–1590 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kotschote M.: Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 679–696 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Lattanzio C., Tzavaras A.E.: Structural properties of stress relaxation and convergence from viscoelasticity to polyconvex elastodynamics. Arch. Ration. Mech. Anal. 180, 449–492 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lattanzio C., Tzavaras A.E.: Relative entropy in diffusive relaxation. SIAM J. Math. Anal. 45, 1563–1584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lattanzio, C., Tzavaras, A.E.: From gas dynamics with large friction to gradient flows describing diffusion theories. (2016) (preprint)

  30. Leger N., Vasseur A.: Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations. Arch. Ration. Mech. Anal. 201, 271–302 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Luo T., Smoller J.: Existence and non-linear stability of rotating star solutions of the compressible Euler–Poisson equations. Arch. Ration. Mech. Anal. 191, 447–496 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, Vol. 17, 2nd edn. Springer, New York 1999

  33. Neusser J., Rohde C., Schleper V.: Relaxation of the Navier–Stokes–Korteweg equations for compressible two-phase flow with phase transition. Int. J. Numer. Method. Fluid. 79, 615–639 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  34. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Peng Y., Wang Y.: Convergence of compressible Euler–Poisson equations to incompressible type Euler equations. Asymp. Anal. 41, 141–160 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Ren X., Truskinovsky L.: Finite scale microstructures in nonlocal elasticity. J. Elasticity 59, 319–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rohde, C.: A local and low-order Navier–Stokes–Korteweg system. In: Nonlinear partial differential equations and hyperbolic wave phenomena, vol. 526 of Contemp. Math., pp. 315–337. Am. Math. Soc. Providence, RI 2010

  38. Solci M., Vitali E.: Variational models for phase separation. Interf. Free Bound. 5, 27–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tian L., Xu Y., Kuerten J.G.M., Vander Vegt J.J.W.: A local discontinuous Galerkin method for the propagation of phase transition in solids and fluids. J. Sci. Comput. 9(3), 688–720 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70563, Stuttgart, Germany

    Jan Giesselmann

  2. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, 67010, Coppito (L’Aquila), AQ, Italy

    Corrado Lattanzio

  3. Computer, Electrical, Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

    Athanasios E. Tzavaras

  4. Institute for Applied and Computational Mathematics, Foundation for Research and Technology, 70013, Heraklion, Crete, Greece

    Athanasios E. Tzavaras

Authors
  1. Jan Giesselmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Corrado Lattanzio
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Athanasios E. Tzavaras
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Athanasios E. Tzavaras.

Additional information

Communicated by F. Otto

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Giesselmann, J., Lattanzio, C. & Tzavaras, A.E. Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics. Arch Rational Mech Anal 223, 1427–1484 (2017). https://doi.org/10.1007/s00205-016-1063-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-1063-2

Search

Navigation