Numbers of equilibria for the equation of self-gravitating isentropic gas surrounding a solid ball
Tóm tắt
For self-gravitating, spherically symmetric and isentropic gas surrounding a solid star, when the adiabatic index γ ∈ [4/3, 2) we prove that there is a unique equilibrium for given momentum and total mass. When γ ∈ (1, 4/3), we prove that there are multiple equilibria for a certain range of momentum and total mass and the number of equilibria may grow arbitrarily larger for certain γ. These results are consistent with the beliefs of astrophisicists; the stationary solutions are stable if γ > 4/3 and unstable if γ < 4/3. The problems were studied through the classical Lane-Emden equation.
Tài liệu tham khảo
S. Chandrasekhar, An Introduction to the Study of Stellar Structures. University of Chicago Press, 1939.
M.K. Kwong and Y. Li., Uniqueness of radial solutions of semilinear elliptic equations. Trans. Amer. Math. Soc.,333 (1992), 339–363.
S.S. Lin, On the existence of positive radial solutions for nonlinear elliptic equations in annular domains. J. Differential Equations,81 (1989), 221–233.
T. Makino, Mathematical Aspects of the Euler-Poisson Equation for the Evolution of Gaseous Stars. Lecture Notes 1993, National Chiao-Tung University, Hsin-chu, Taiwan, March 1993.
T. Makino, On the spherically symmetric motion of self-gravitating isentropic gas surrounding a solid ball. Preprint.
W.-M. Ni and R. Nussbaum, Uniqueness and non-uniqueness for positive radial solution of Δu+f(u,r)=0. Comm. Pure Appl. Math.,38 (1985), 67–108.
W.-M. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case. Accad. Naz. Lincei,77 (1986), 231–257.
N. Straumann, General Relativity and Relativistic Astrophysics. Springer-Verlag, 1991.