×

On the stability of a stellar structure in one dimension. II: The reactive case. (English) Zbl 0882.76025

Summary: We complete in this paper [for part I, see the author, Math. Models Methods Appl. Sci. 6, No. 3, 365-383 (1996; Zbl 0853.76075)] the study of stability of the interface in a free boundary problem for a self-gravitating gas in one space dimension, with an external pressure \(P\), and a Fourier coefficient \(\lambda\), for the thermal flux, including a chemical, self-consistent, reacting process. In the non-radiative limit, we find different possible asymptotic behaviours: if \(\lambda>0\), the gas tends to collapse; if \(\lambda=0\), we show that, when \(P>0\), the solution converges for large time to the isothermal solution of the corresponding stationary problem, while for \(P=0\), under some additional condition connecting the total energy and the mass of the structure, the system is unstable, and the gas tends to fill the space. In the limit of the photon gas, we show that analogous asymptotics hold.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
76V05 Reaction effects in flows
85A15 Galactic and stellar structure
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 0853.76075
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] GUI-QIANG CHEN, 1992, Global solution to the compressible Navier-Stokes equations for a reacting mixture, SIAM J. Math. Anal. 23, 609-634. Zbl0771.35044 MR1158824 · Zbl 0771.35044 · doi:10.1137/0523031
[2] S. F. SHANDARIN, Ya. B. ZELDOVITCH, 1989, The large-scale structure of the universe : Turbulence, intermittency, structures in a self-gravitating medium, Reviews of Modern Physics, 61, 185-220. MR989562
[3] F. WILLIAMS, 1983, Combustion theory, Addison-Wesley. · Zbl 0516.76104
[4] Y. B. ZELDOVITCH, Yu. RAISER, 1966, Physics of shock waves and high temperature hydrodynamic phenomena, Academic Press.
[5] R. KlPPENHAHN, A. WEINGERT, 1994, Stellar structure and évolution, Springer Verlag.
[6] D. KINDERLEHRER, G. STAMPACCHIA, 1980, An introduction to variational inequalities and their applications, Academic Press. Zbl0457.35001 MR567696 · Zbl 0457.35001
[7] T. NAGASAWA, 1988, On the outer pressure problem of the one-dimensional polytropic ideal gas, Japan J. Appl. Math., 5, 53-85. Zbl0665.76076 MR924744 · Zbl 0665.76076 · doi:10.1007/BF03167901
[8] V.A. SOLONNIKOV, A. V. KAZHIKHOV, 1981, Existence theorems for the equations of motion of a compressible viscous fluid, Ann. Rev. Fluid Mech., 13, 79-95. Zbl0492.76074 · Zbl 0492.76074
[9] S. N. ANTONSEV, A.V. KAZHIKOV, V. N. MONAKHOV, 1990, Boundary value problems in mechanics of non-homogeneous fluids, North Holland. Zbl0696.76001 · Zbl 0696.76001
[10] P. BLOTTIAU, S. BOUQUET, J. P. CHIÈZE, 1988, An asymptotic self-similar solution for the gravitational collapse, Astron. Astrophys., 207, 24-36.
[11] T. MAKINO, B. PERTHAME, 1990, Sur les solutions à symétrie sphériques de l’équation d’Euler-Poisson pour l’évolution d’étoiles gazeuses, Japan J. Appl Math., 7, 165-170. Zbl0743.35048 MR1039243 · Zbl 0743.35048 · doi:10.1007/BF03167897
[12] B. DUCOMET, 1995, Evolution of a self-gravitating Rosseland gas in one dimension, Math. Models and Methods in Appl. Sci, 5, 999-1012. Zbl0838.76079 MR1359217 · Zbl 0838.76079 · doi:10.1142/S0218202595000528
[13] B. DUCOMET, 1996, On the stability of a stellar structure in one dimension, Math. Models and Methods in Appl. Sci., 6, 365-383. Zbl0853.76075 MR1388712 · Zbl 0853.76075 · doi:10.1142/S0218202596000134
[14] T. NAGASAWA, 1988, On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with stress-free condition, Quart. of Appl. Math., 46, 665-679. Zbl0693.76075 MR973382 · Zbl 0693.76075
[15] D. HOFF, 1987, Global existence for ld, compressible isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc, 303, 169-181. Zbl0656.76064 MR896014 · Zbl 0656.76064 · doi:10.2307/2000785
[16] D. HOFF, 1991, Discontinuous solutions of the Navier-Stokes for compressible flows, Arch. Rational Mech. Anal., 114, 15-46. Zbl0732.35071 MR1088275 · Zbl 0732.35071 · doi:10.1007/BF00375683
[17] D. HOFF, 1992, Global well-posedness of the Cauchy problem for the Navier-Stokes equations of non-isentropic flow with discontinuous initial data, Journal of Diff. Equ., 95,33-74. Zbl0762.35085 MR1142276 · Zbl 0762.35085 · doi:10.1016/0022-0396(92)90042-L
[18] D. SERRE, 1986, Solutions faible globales des équations de Navier-Stokes pour un fluide compressible, C.R. Acad. Sci. Paris, 303, Série I, 639-642. Zbl0597.76067 MR867555 · Zbl 0597.76067
[19] D. SERRE, 1986, Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de la chaleur, C.R. Acad. Sci. Paris, 303, Série I, 703-706. Zbl0611.35070 MR870700 · Zbl 0611.35070
[20] D. SERRE, 1991, Évolution d’une masse finie de fluide en dimension 1, In : Nonlinear Partial Differential Equations, H. Brézis et J. L. Lions Ed., Longman. Zbl0729.76027 · Zbl 0729.76027
[21] S. CHANDRASEKHAR, 1957, An Introduction to the Study of Stellar Structure, Dover. Zbl0079.23901 MR92663 · Zbl 0079.23901
[22] B. DUCOMET, Hydrodynamical models of stars, To appear in Reviews in Mathematical Physics.
[23] M. OKADA, T. MAKINO, 1993, Free boundary problem for the equation of spherically symmetric motion of viscous gas, Japan J. Indust. Appl. Math., 10, 219-235. Zbl0783.76082 MR1227730 · Zbl 0783.76082 · doi:10.1007/BF03167573
[24] K. MIZOHATA, 1993, Global weak solutions for the equation of isothermal gas around a star, Preprint, Dep. of Inform. Sci., Tokyo Inst. of Technology. Zbl0833.35114 MR1295943 · Zbl 0833.35114
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.