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On the expanding configurations of viscous radiation gaseous stars: thermodynamic model. (English) Zbl 1433.35410

Summary: In this work, we study the stability of the expanding configurations of radiation gaseous stars. Such expanding configurations exist for a thermodynamic model, given as a class of self-similar solutions to the associated dynamic system with viscosity coefficients satisfying \(2 \mu + 3 \lambda = 0\) for the monatomic gas; that is, the bulk viscosity is vanishing. With respect to small perturbations, this work shows that the linearly expanding homogeneous solutions are stable for a large expanding rate. This is an extensive study of the result in [M. Hadžić and J. Jang, Commun. Pure Appl. Math. 71, No. 5, 827–891 (2018; Zbl 1390.35246)].

MSC:

35Q85 PDEs in connection with astronomy and astrophysics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76E20 Stability and instability of geophysical and astrophysical flows
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35R37 Moving boundary problems for PDEs
35C06 Self-similar solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics
85A25 Radiative transfer in astronomy and astrophysics

Citations:

Zbl 1390.35246
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References:

[1] Auchmuty, J. F.G.; Beals, Richard, Variational solutions of some nonlinear free boundary problems, Arch. Ration. Mech. Anal., 43, 4, 255-271 (1971) · Zbl 0225.49013
[2] Caffarelli, Luis A.; Friedman, Avner, The shape of axisymmetric rotating fluid, J. Funct. Anal., 35, 1, 109-142 (1980) · Zbl 0439.35068
[3] Chandrasekhar, S., An Introduction to the Study of Stellar Structure (1958), Dover Publications, Inc. · Zbl 0079.23901
[4] Chanillo, Sagun; Li, Yan Yan, On diameters of uniformly rotating stars, Commun. Math. Phys., 166, 2, 417-430 (1994) · Zbl 0816.76076
[5] Chanillo, Sagun; Weiss, Georg S., A remark on the geometry of uniformly rotating stars, J. Differ. Equ., 253, 2, 553-562 (2012) · Zbl 1245.85001
[6] Coutand, Daniel; Lindblad, Hans; Shkoller, Steve, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Commun. Math. Phys., 296, 2, 559-587 (2010) · Zbl 1193.35139
[7] Coutand, Daniel; Shkoller, Steve, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Commun. Pure Appl. Math., LXIV, 328-366 (2011) · Zbl 1217.35119
[8] Coutand, Daniel; Shkoller, Steve, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206, 2, 515-616 (2012) · Zbl 1257.35147
[9] Deng, Yinbin; Liu, Tai-Ping; Yang, Tong; Yao, Zheng-an, Solutions of Euler-Poisson equations for gaseous stars, Arch. Ration. Mech. Anal., 164, 3, 261-285 (2002) · Zbl 1038.76036
[10] Federbush, Paul; Luo, Tao; Smoller, Joel, Existence of magnetic compressible fluid stars, Arch. Ration. Mech. Anal., 215, 2, 611-631 (2014) · Zbl 1308.35206
[11] Friedman, Avner; Turkington, Bruce, Existence and dimensions of a rotating white dwarf, J. Differ. Equ., 42, 414-437 (1981) · Zbl 0493.76109
[12] Fu, Chun-chieh; Lin, Song-Sun, On the critical mass of the collapse of a gaseous star in spherically symmetric and isentropic motion, Jpn. J. Ind. Appl. Math., 15, 461-469 (1998) · Zbl 0913.35108
[13] Goldreich, Peter; Weber, Stephen V., Homologously collapsing stellar cores, Astrophys. J., Am. Astron. Soc., 238, 1, 991-997 (1980)
[14] Hadžić, Mahir; Jang, Juhi, Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system, Commun. Pure Appl. Math., 71, 5, 1-46 (2017) · Zbl 1390.35246
[15] Jang, Juhi, Nonlinear instability in gravitational Euler-Poisson systems for \(\gamma = \frac{6}{5} \), Arch. Ration. Mech. Anal., 188, 265-307 (2008) · Zbl 1192.85003
[16] Jang, Juhi, Local well-posedness of dynamics of viscous gaseous stars, Arch. Ration. Mech. Anal., 195, 3, 797-863 (2010) · Zbl 1197.35294
[17] Jang, Juhi, Nonlinear instability theory of Lane-Emden stars, Commun. Pure Appl. Math., 67, 9, 1418-1465 (2014) · Zbl 1309.35080
[18] Jang, Juhi; Masmoudi, Nader, Well-posedness for compressible Euler equations with physical vacuum singularity, Commun. Pure Appl. Math., LXII, 1327-1385 (2009) · Zbl 1213.35298
[19] Jang, Juhi; Masmoudi, Nader, Well-posedness of compressible Euler equations in a physical vacuum, Commun. Pure Appl. Math., LXVIII, Article 0061 pp. (2015) · Zbl 1317.35185
[20] Jang, Juhi; Tice, Ian, Instability theory of the Navier-Stokes-Poisson equations, Anal. PDE, 6, 5, 1121-1181 (2013) · Zbl 1284.35317
[21] Li, YanYan, On uniformly rotating stars, Arch. Ration. Mech. Anal., 115, 4, 367-393 (1991) · Zbl 0850.76784
[22] Lieb, Elliott H.; Yau, Horng-tzer, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112, 147-174 (1987) · Zbl 0641.35065
[23] Lin, Song-Sun, Stability of gaseous stars in spherically symmetric motions, SIAM J. Math. Anal., 28, 3, 539-569 (1997) · Zbl 0871.35012
[24] Lions, Pierre-Louis, Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models, Oxford Lecture Series in Mathematics and Its Applications, vol. 3 (1996), Oxford University Press · Zbl 0866.76002
[25] Liu, Tai-Ping, Compressible flow with damping and vacuum, Jpn. J. Ind. Appl. Math., 13, 25-32 (1996) · Zbl 0865.35107
[26] Liu, Xin, A model of radiational gaseous stars (2016) · Zbl 1423.76392
[27] Luo, Tao; Smoller, Joel, Rotating fluids with self-gravitation in bounded domains, Arch. Ration. Mech. Anal., 173, 3, 345-377 (2004) · Zbl 1060.76125
[28] Luo, Tao; Smoller, Joel, Nonlinear dynamical stability of Newtonian rotating and non-rotating white dwarfs and rotating supermassive stars, Commun. Math. Phys., 284, 2, 425-457 (2008) · Zbl 1166.35031
[29] Luo, Tao; Smoller, Joel, Existence and non-linear stability of rotating star solutions of the compressible Euler-Poisson equations, Arch. Ration. Mech. Anal., 191, 447-496 (2009) · Zbl 1163.85001
[30] Luo, Tao; Xin, Zhouping; Zeng, Huihui, 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, 3, 763-831 (2014) · Zbl 1309.35065
[31] Luo, Tao; Xin, Zhouping; Zeng, Huihui, Nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem with degenerate density dependent viscosities, Commun. Math. Phys., 347, 3, 657-702 (2016) · Zbl 1351.35227
[32] Luo, Tao; Xin, Zhouping; Zeng, Huihui, On nonlinear asymptotic stability of the Lane-Emden solutions for the viscous gaseous star problem, Adv. Math. (N. Y.), 291, 90-182 (2016) · Zbl 1344.35150
[33] Luo, Tao; Zeng, Huihui, Global existence of smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping, Commun. Pure Appl. Math. (2015) · Zbl 1344.35086
[34] Rein, Gerhard, Non-linear stability of gaseous stars, Arch. Ration. Mech. Anal., 168, 2, 115-130 (2003) · Zbl 1044.76026
[35] Wu, Yilun, On rotating star solutions to the non-isentropic Euler-Poisson equations, J. Differ. Equ., 259, 12, 7161-7198 (2015) · Zbl 1333.35173
[36] Zeng, Huihui, Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier-Stokes equations, Nonlinearity, 28, 2, 331-345 (2015) · Zbl 1323.35222
[37] Zeng, Huihui, Global resolution of the physical vacuum singularity for three-dimensional isentropic inviscid flows with damping in spherically symmetric motions, Arch. Ration. Mech. Anal., 226, 1, 33-82 (2017) · Zbl 1383.35150
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