On the global existence for the compressible Euler–Poisson system, and the instability of static solutions


We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index \(\gamma \) of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Sér I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field \(u_0\) such that the spectrum of \(Du_0\) is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

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  1. 1.

    There is no upper bound for s if \(\gamma =1+2/k\) for some integer \(k\ge 2,\) or if \(\kappa =0\).

  2. 2.

    Second step actually fails in the two-dimensional case, owing to the ‘bad’ properties of the homogeneous Sobolev space \(\dot{H}^1(\mathbb {R}^2).\) This is the reason why we focus throughout on the case \(d\ge 3\).


  1. 1.

    H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, 343, Springer (2011).

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    M. Bézard. Existence locale de solutions pour les équations d’Euler-Poisson. Japan J. Indust. Appl. Math., 10:431–450, 1993.

    MathSciNet  Article  Google Scholar 

  4. 4.

    X. Blanc, R. Danchin, B. Ducomet, Š. Nečasova. The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling. J. Hyperbolic Differ. Equ.arXiv:1906.08075.

  5. 5.

    U. Brauer and L. Karp. Local existence of solutions to the Euler-Poisson system including densities without compact support. Journal of Differential Equations, 264:755–785, 2018.

    MathSciNet  Article  Google Scholar 

  6. 6.

    S. Chandrasekhar. An introduction to the study of stellar structure. Dover Publications, New York, 1957.

    Google Scholar 

  7. 7.

    S.G. Chefranov and A.S. Chefranov. Exact time-dependent solution to the three-dimensional Euler-Helmholtz and Riemann-Hopf equations for vortex flow of a compressible medium and one of the millenium prize problems. arXiv:1703.07239v3.

  8. 8.

    J.-Y. Chemin. Dynamique des gaz à masse totale finie. Asymptotic Analysis, 3:215–220, 1990.

    MathSciNet  Article  Google Scholar 

  9. 9.

    G.Q. Chen and D. Wang. The Cauchy problem for the Euler equations for compressible fluids. in “Handbook of Mathematical Fluid Dynamics, Vol. 1”, S. Friedlander, D. Serre Eds. North-Holland, Elsevier, Amsterdam, Boston, London, New York, 2002.

  10. 10.

    H-Y. Chiu. Stellar physics. Blaisdell Publishing Company, Waltham, Toronto, London, 1968.

  11. 11.

    P. Gamblin. Solution régulière à temps petit pour l’équation d’Euler-Poisson. Commun. in Partial Differential Equations, 18:731–745, 1993.

    MathSciNet  Article  Google Scholar 

  12. 12.

    B. Gidas, W-M. Ni and L. Nirenberg. Symmetry and related properties via the maximum principle. Commun. Math. Phys., 68:209–243, 1979.

    MathSciNet  Article  Google Scholar 

  13. 13.

    M. Grassin and D. Serre. Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique. C.R. Acad. Sci. Paris, Série I, 325:721–726, 1997.

  14. 14.

    M. Grassin. Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 47:1397–1432, 1998.

    MathSciNet  Article  Google Scholar 

  15. 15.

    C. Gu, Z. Lei. Local well-posedness of the three dimensional compressible Euler-Poisson equations with physical vacuum. J. Math. Pures Appl., 105(9), no. 5, 662–723, 2016.

  16. 16.

    J. Jang. Nonlinear instability in gravitational Euler-Poisson system for \(\gamma =6/5\). Arch. Ration. Mech. Anal., 188:265–307, 2008.

    MathSciNet  Article  Google Scholar 

  17. 17.

    D. Kateb. On the boundedness of the mapping \(f\mapsto |f|^\mu ,\)\(\mu >1\) on Besov spaces. Math. Nachr., 248/249:110–128 (2003).

    MathSciNet  Article  Google Scholar 

  18. 18.

    T. Kato. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal., 58:181–205, 1975.

    MathSciNet  Article  Google Scholar 

  19. 19.

    M. Lécureux-Mercier. Global smooth solutions of Euler equations for Van der Waals gases. SIAM J. Math. Anal., 43:877–903, 2011.

    MathSciNet  Article  Google Scholar 

  20. 20.

    D. Li. On Kato-Ponce and fractional Leibniz. Revista Matematica Iberoamericana, 35(1):23–100, 2019.

    MathSciNet  Article  Google Scholar 

  21. 21.

    S-S. Lin. Stability of gaseous stars in spherically symmetric motions. SIAM J. Math. Anal., 28:539–569, 1997.

  22. 22.

    T. Luo, Z. Xin, H. Zeng. Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation. Arch. Ration. Mech. Anal., 213:763–831, 2014.

    MathSciNet  Article  Google Scholar 

  23. 23.

    A. Majda. Compressible fluid flow and systems of conservation laws in several variables. Springer-Verlag, New-York, Berlin, Heidelberg, Tokyo, 1984.

    Google Scholar 

  24. 24.

    T. Makino. On a local existence theorem for the evolution equation of gaseous stars. In Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations, 3:459–479, 1986.

    MathSciNet  Article  Google Scholar 

  25. 25.

    T. Makino. Blowing-up solutions of the Euler-Poisson equations for the evolution of gaseous stars. Transport Theory and Statistical Physics, 21:615–624, 1992.

    MathSciNet  Article  Google Scholar 

  26. 26.

    T. Makino. Mathematical aspects of the Euler-Poisson equations for the evolution of gaseous stars. NCTU-MATH 930001, Lect. Notes Dep. of Applied Math., National Chiao Tung University, Taiwan, R.O.C., March 2003.

  27. 27.

    T. Makino and B. Perthame. Sur les solutions à symétrie sphérique de l’équation d’Euler-Poisson pour l’évolution d’étoiles gazeuses. Japan J. Appl. Math., 7:165–170, 1990.

    MathSciNet  Article  Google Scholar 

  28. 28.

    T. Makino and S. Ukai. Sur l’existence des solutions locales de l’équation d’Euler-Poisson pour l’évolution d’étoiles gazeuses. J. Math. Kyoto Univ, 27:387–399, 1987.

    MathSciNet  Article  Google Scholar 

  29. 29.

    B. Perthame. Non-existence of global solutions to Euler-Poisson equations for repulsive forces. Japan J. Appl. Math., 7:363–367, 1990.

    MathSciNet  Article  Google Scholar 

  30. 30.

    G. Rein. Nonlinear stability of gaseous stars. Arch. Ration. Mech. Anal., 168:115–130, 2003.

    MathSciNet  Article  Google Scholar 

  31. 31.

    E. Schatzman and F. Praderie. Les étoiles. InterEditions, Editions du CNRS, Paris, 1990.

    Google Scholar 

  32. 32.

    D. Serre. Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, Grenoble, 47:139–159, 1997.

  33. 33.

    T.C. Sideris. Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum. Arch. Ration. Mech. Anal., 225:141–176, 2017.

    MathSciNet  Article  Google Scholar 

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The authors have been partially supported by the ANR project INFAMIE (ANR-15-CE40-0011).

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For the reader’s convenience, we here recall some technical results that have been used in the paper. More details may be found in the appendix of [4].

The first result is a differential inequality that leads to a control of the solution for all positive times.

Lemma 3.1

Let \(Y:\mathbb {R}_+\rightarrow \mathbb {R}_+\) satisfy the differential inequality

$$\begin{aligned} \frac{\text {d}}{{\text {d}}t}Y+\frac{a}{1+t}Y\le C\biggl (\frac{Y}{(1+t)^2}+Y^2+(1+t)^{m'-1}Y^{m+1}\biggr )\quad \hbox {on }\ \mathbb {R}_+ \end{aligned}$$

for some \(C>0,\) \(a>1,\) \(m>0\) and \(m'<ma.\) Then, there exists \(c=c(a,m,m',C)\) such that if \(Y(0)\le c,\) then we have

$$\begin{aligned} Y(t)\le 2e^{\frac{Ct}{1+t}}\frac{Y_0}{(1+t)^a}\quad \hbox {for all }\ t\ge 0. \end{aligned}$$

The following result (see [4, 17]) has been used to bound the potential term.

Lemma 3.2

Let \(\alpha \ge 1\) and \(0\le \sigma <\alpha +\frac{1}{2}\). Then, we have the inequality

$$\begin{aligned} \Vert |z|^\alpha \Vert _{\dot{H}^\sigma }\lesssim \Vert z\Vert _{L^\infty }^{\alpha -1}\Vert z\Vert _{\dot{H}^\sigma }. \end{aligned}$$

We also used the following first-order commutator estimate that corresponds to the end of [20, Rem. 1.5], or may be seen as a straightforward adaptation to the homogeneous framework of the second inequality of [2, Lemma A.2]:

Lemma 3.3

If \(s>0,\) then we have:

$$\begin{aligned} \Vert [v,\dot{\Lambda }^s]u\Vert _{L^2}\lesssim \Vert v\Vert _{\dot{H}^s}\Vert u\Vert _{L^\infty }+\Vert \nabla v\Vert _{L^\infty }\Vert u\Vert _{\dot{H}^{s-1}}. \end{aligned}$$

The following second-order commutator inequality that has been established in [4] (after a prior result in [2]) enabled us to prove the \(\dot{H}^s\) estimate for the solutions to (3.2).

Lemma 3.4

If \(s>1\) then, we have:

$$\begin{aligned} \Vert [v,\dot{\Lambda }^s]u-s\nabla v\cdot \dot{\Lambda }^{s-2}\nabla u\Vert _{L^2}\lesssim \Vert v\Vert _{\dot{H}^s}\Vert u\Vert _{L^\infty }+\Vert \nabla ^2v\Vert _{L^\infty }\Vert u\Vert _{\dot{H}^{s-2}}. \end{aligned}$$

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Danchin, R., Ducomet, B. On the global existence for the compressible Euler–Poisson system, and the instability of static solutions. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00639-1

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Mathematics Subject Classification

  • 35Q30
  • 76N10


  • Compressible Euler system
  • Poisson
  • Global solution
  • Decay
  • Instability