## On the global existence for the compressible Euler-Poisson system, and the instability of static solutions.(English)Zbl 1481.35322

Summary: 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 [M. Grassin and D. Serre, C. R. Acad. Sci., Paris, Sér. I, Math. 325, No. 7, 721–726 (1997; Zbl 0887.35125); M. Grassin, Indiana Univ. Math. J. 47, No. 4, 1397–1432 (1998; Zbl 0930.35134)], 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 [X. Blanc, the authors and Š. Nečasová, J. Hyperbolic Differ. Equ. 18, No. 1, 169–193 (2021; Zbl 1473.35396)], we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

### Citations:

Zbl 0887.35125; Zbl 0930.35134; Zbl 1473.35396
Full Text:

### References:

 [1] H. Bahouri, J.-Y. Chemin and R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, 343, Springer (2011). · Zbl 1227.35004 [2] 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, 4, 1499-1579 (2007) · Zbl 1125.76060 [3] Bézard, M., Existence locale de solutions pour les équations d’Euler-Poisson, Japan J. Indust. Appl. Math., 10, 431-450 (1993) · Zbl 0811.35095 [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. · Zbl 1473.35396 [5] Brauer, U.; Karp, L., Local existence of solutions to the Euler-Poisson system including densities without compact support, Journal of Differential Equations, 264, 755-785 (2018) · Zbl 1378.35302 [6] Chandrasekhar, S., An introduction to the study of stellar structure (1957), New York: Dover Publications, New York · Zbl 0079.23901 [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. · Zbl 1453.35136 [8] Chemin, J-Y, Dynamique des gaz à masse totale finie, Asymptotic Analysis, 3, 215-220 (1990) · Zbl 0708.76110 [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. · Zbl 1230.35096 [10] H-Y. Chiu. Stellar physics. Blaisdell Publishing Company, Waltham, Toronto, London, 1968. [11] Gamblin, P., Solution régulière à temps petit pour l’équation d’Euler-Poisson, Commun. in Partial Differential Equations, 18, 731-745 (1993) · Zbl 0782.35058 [12] Gidas, B.; Ni, W-M; Nirenberg, L., Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 [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. · Zbl 0887.35125 [14] Grassin, M., Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J., 47, 1397-1432 (1998) · Zbl 0930.35134 [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. · Zbl 1346.35159 [16] Jang, J., Nonlinear instability in gravitational Euler-Poisson system for $$\gamma =6/5$$, Arch. Ration. Mech. Anal., 188, 265-307 (2008) · Zbl 1192.85003 [17] Kateb, D., On the boundedness of the mapping $$f\mapsto |f|^\mu,\mu >1$$ on Besov spaces, Math. Nachr., 248, 249, 110-128 (2003) · Zbl 1015.46022 [18] Kato, T., The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58, 181-205 (1975) · Zbl 0343.35056 [19] Lécureux-Mercier, M., Global smooth solutions of Euler equations for Van der Waals gases, SIAM J. Math. Anal., 43, 877-903 (2011) · Zbl 1230.35063 [20] Li, D., On Kato-Ponce and fractional Leibniz, Revista Matematica Iberoamericana, 35, 1, 23-100 (2019) · Zbl 1412.35261 [21] S-S. Lin. Stability of gaseous stars in spherically symmetric motions. SIAM J. Math. Anal., 28:539-569, 1997. · Zbl 0871.35012 [22] Luo, T.; Xin, Z.; Zeng, H., 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) · Zbl 1309.35065 [23] Majda, A., Compressible fluid flow and systems of conservation laws in several variables (1984), New-York, Berlin, Heidelberg, Tokyo: Springer-Verlag, New-York, Berlin, Heidelberg, Tokyo · Zbl 0537.76001 [24] Makino, T., On a local existence theorem for the evolution equation of gaseous stars, Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations, 3, 459-479 (1986) [25] Makino, T., Blowing-up solutions of the Euler-Poisson equations for the evolution of gaseous stars, Transport Theory and Statistical Physics, 21, 615-624 (1992) · Zbl 0793.76069 [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] Makino, T.; Perthame, B., 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) · Zbl 0743.35048 [28] Makino, T.; Ukai, S., 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) · Zbl 0657.35113 [29] Perthame, B., Non-existence of global solutions to Euler-Poisson equations for repulsive forces, Japan J. Appl. Math., 7, 363-367 (1990) · Zbl 0717.35049 [30] Rein, G., Nonlinear stability of gaseous stars, Arch. Ration. Mech. Anal., 168, 115-130 (2003) · Zbl 1044.76026 [31] Schatzman, E.; Praderie, F., Les étoiles (1990), Editions du CNRS, Paris: InterEditions, Editions du CNRS, Paris [32] D. Serre. Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier, Grenoble, 47:139-159, 1997. · Zbl 0864.35069 [33] Sideris, TC, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal., 225, 141-176 (2017) · Zbl 1367.35115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.