On Global Well/Ill-Posedness of the Euler-Poisson System

  • Eduard Feireisl
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


We discuss the problem of well-posedness of the Euler-Poisson system arising, for example, in the theory of semi-conductors, models of plasma and gaseous stars in astrophysics. We introduce the concept of dissipative weak solution satisfying, in addition to the standard system of integral identities replacing the original system of partial differential equations, the balance of total energy, together with the associated relative entropy inequality. We show that strong solutions are unique in the class of dissipative solutions (weak-strong uniqueness). Finally, we use the method of convex integration to show that the Euler-Poisson system may admit even infinitely many weak dissipative solutions emanating from the same initial data.


Dissipative solution Euler-Poisson system Weak solution 



The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078.


  1. 1.
    T. Alazard, Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (2006) (electronic)Google Scholar
  2. 2.
    F. Berthelin, A. Vasseur, From kinetic equations to multidimensional isentropic gas dynamics before shocks. SIAM J. Math. Anal. 36, 1807–1835 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    E. Chiodaroli, A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11(3), 493–519 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    E. Chiodaroli, E. Feireisl, O. Kreml, On the weak solutions to the equations of a compressible heat conducting gas. Ann. Inst. Henri Poincaré Anal Non Linéaire. 32(1), 225–243 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    C.M. Dafermos, The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    C. De Lellis, L. Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    E. Feireisl, Relative entropies in thermodynamics of complete fluid systems. Discrete Cont. Dyn. Syst. Ser. A 32, 3059–3080 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    E. Feireisl, A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    P. Germain, Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Y. Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R 3+1. Commun. Math. Phys. 195, 249–265 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Y. Guo, B. Pausader, Global smooth ion dynamics in the Euler-Poisson system. Commun. Math. Phys. 308, 89–125 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Y. Guo, A.S. Tahvildar-Zadeh, Nonlinear partial differential equations, in Contemporary Mathematics (American Mathematical Society, Providence, RI, 1999), pp. 151–161CrossRefMATHGoogle Scholar
  13. 13.
    A. Jüengel, Transport Equations for Semiconductors. Lecture Notes in Physis, vol. 773 (Springer, Heidelberg, 2009)Google Scholar
  14. 14.
    P.-L. Lions, Mathematical Topics in Fluid Dynamics: Vol. 1, Incompressible models (Oxford Science Publication, Oxford, 1996)Google Scholar
  15. 15.
    D. Serre, Local existence for viscous system of conservation laws: H s-data with s > 1 + d∕2, in Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, vol. 526. Contemporary Mathematics (American Mathematical Society, Providence, RI, 2010), pp. 339–358Google Scholar
  16. 16.
    D. Serre, The structure of dissipative viscous system of conservation laws. Phys. D 239(15), 1381–1386 (2010)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Basel 2016

Authors and Affiliations

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

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