Weak Solutions to Problems Involving Inviscid Fluids

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 183)

Abstract

We consider an abstract functional-differential equation derived from the pressureless Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy.

Keywords

Euler system Weak solution Convex integration 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Antonelli, P., Marcati, P.: The quantum hydrodynamics system in two space dimensions. Arch. Rational Mech. Anal. 203, 499–527 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Audiard, C.: Dispersive smoothing for the Euler-Korteweg model. SIAM J. Math. Anal. 44(4), 3018–3040 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benzoni-Gavage, S.: Spectral transverse instability of solitary waves in Korteweg fluids. J. Math. Anal. Appl. 361(2), 338–357 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Benzoni-Gavage, S.: Planar traveling waves in capillary fluids. Differ. Integral Equ. 26(3–4), 439–485 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Bresch, D., Desjardins, B., Ducomet, B.: Quasi-neutral limit for a viscous capillary model of plasma. Ann. Inst. Poincare 22, 1–9 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chiodaroli, E.: A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11(3), 493–519 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chiodaroli, E., Feireisl, E., Kreml, O.: On the weak solutions to the equations of a compressible heat conducting gas. Annal. Inst. Poincaré, Anal. Nonlinear 32, 225–243 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  10. 10.
    Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70, 167–179 (1979)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    De Lellis, C., Székelyhidi Jr., L.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Donatelli, D., Feireisl, E., Marcati, P.: Well/ill posedness for the Euler-Korteweg-poisson system and related problems. Commun. Partial Differ. Equ. 40(7), 1314–1335 (2015)Google Scholar
  14. 14.
    Feireisl, E.: On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discret. Contin. Dyn. S.—Ser. S. 9(1), 173–183 (2014)Google Scholar
  15. 15.
    Jüngel, A.: Transport Equations for Semiconductors. Lecture Notes in Physics, vol. 773. Springer, Berlin (2009)Google Scholar
  16. 16.
    Kotschote, M.: Strong well-posedness for a Korteweg-type model for the dynamics of a compressible non-isothermal fluid. J. Math. Fluid Mech. 12(4), 473–484 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kotschote, M.: Dynamics of compressible non-isothermal fluids of non-Newtonian Korteweg type. SIAM J. Math. Anal. 44(1), 74–101 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Knopps, L.J. (ed.) Nonlinear Analysis and Mechanics, Heriot-Watt Symposium. Research Notes in Math, vol. 39, pp. 136–211. Pitman, Boston (1975)Google Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic

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