Global continuation and the theory of rotating stars
Author:
Yilun Wu
Journal:
Quart. Appl. Math. 78 (2020), 147-159
MSC (2010):
Primary 35Q35
DOI:
https://doi.org/10.1090/qam/1550
Published electronically:
July 19, 2019
MathSciNet review:
4042222
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Abstract: This paper gives a condensed review of the history of solutions to the Euler-Poisson equations modeling equilibrium states of rotating stars and galaxies, leading to a recent result of Walter Strauss and the author. This result constructs a connected set of rotating star solutions for larger and larger rotation speed, so that the supports of the stars become unbounded if we assume an equation of state $p = \rho ^\gamma$, $4/3<\gamma <2$. On the other hand, if $6/5<\gamma <4/3$, we show that either the supports of the stars become unbounded, or the density somewhere within the stars becomes unbounded. This is the first global continuation result for rotating stars that displays singularity formation within the solution set.
References
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References
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- Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI https://doi.org/10.1016/0022-1236%2873%2990051-7
- J. F. G. Auchmuty and Richard Beals, Variational solutions of some nonlinear free boundary problems, Arch. Rational Mech. Anal. 43 (1971), 255–271. MR 0337260, DOI https://doi.org/10.1007/BF00250465
- Luis A. Caffarelli and Avner Friedman, The shape of axisymmetric rotating fluid, J. Funct. Anal. 35 (1980), no. 1, 109–142. MR 560219, DOI https://doi.org/10.1016/0022-1236%2880%2990082-8
- S. Chandrasekhar, An introduction to the study of stellar structure, University of Chicago Press, Chicago, 1939.
- S. Chandrasekhar, Ellipsoidal figures of equilibrium—an historical account, Comm. Pure Appl. Math. 20 (1967), 251–265. MR 0213075, DOI https://doi.org/10.1002/cpa.3160200203
- Sagun Chanillo and Yan Yan Li, On diameters of uniformly rotating stars, Comm. Math. Phys. 166 (1994), no. 2, 417–430. MR 1309557
- D. G. de Figueiredo, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), no. 1, 41–63. MR 664341
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- Leon Lichtenstein, Untersuchungen über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen, Math. Z. 36 (1933), no. 1, 481–562 (German). MR 1545356, DOI https://doi.org/10.1007/BF01188634
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- Walter A. Strauss and Yilun Wu, Rapidly Rotating Stars, Comm. Math. Phys. 368 (2019), no. 2, 701–721. MR 3949722, DOI https://doi.org/10.1007/s00220-019-03414-7
- Yilun Wu, On rotating star solutions to the non-isentropic Euler-Poisson equations, J. Differential Equations 259 (2015), no. 12, 7161–7198. MR 3401594, DOI https://doi.org/10.1016/j.jde.2015.08.016
- Yilun Wu, Existence of rotating planet solutions to the Euler-Poisson equations with an inner hard core, Arch. Ration. Mech. Anal. 219 (2016), no. 1, 1–26. MR 3437846, DOI https://doi.org/10.1007/s00205-015-0891-9
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Additional Information
Yilun Wu
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
MR Author ID:
1127092
Email:
allenwu@ou.edu
Received by editor(s):
April 9, 2019
Received by editor(s) in revised form:
May 28, 2019
Published electronically:
July 19, 2019
Article copyright:
© Copyright 2019
Brown University