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On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary

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Abstract

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler–Poisson system in moving bounded simply-connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data that are small perturbations of size \({\epsilon}\) of these static states, measured in appropriate Sobolev spaces, the solution exists and the perturbation remains of size \({\epsilon}\) on a time interval of length at least \({c\epsilon^{-2},}\) where c is a constant independent of \({\epsilon.}\) This should be compared with the lifespan \({O(\epsilon^{-1})}\) provided by local well-posedness. The key ingredient of our proof is finding a two-step nonlinear transformation which removes quadratic terms from the nonlinearity. Compared with the gravity water wave problem, besides the different geometry of the bounded moving domain, an important difference is that the gravity in water waves is a constant vector, while the self-gravity in the Euler–Poisson system depends nonlinearly on the interface.

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Authors and Affiliations

  1. Department of Mathematics, The University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Lydia Bieri, Shuang Miao, Sohrab Shahshahani & Sijue Wu

Authors
  1. Lydia Bieri
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  2. Shuang Miao
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  3. Sohrab Shahshahani
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  4. Sijue Wu
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Corresponding author

Correspondence to Sijue Wu.

Additional information

Communicated by H.-T. Yau

Support of the National Science Foundation grants DMS-1253149 for the first and second, NSF-1045119 for the third, and DMS-1361791 for the fourth authors is gratefully acknowledged. The third author was also supported by the NSF under Grant No.0932078000 while in residence at the MSRI in Berkeley, CA during Fall 2015.

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Bieri, L., Miao, S., Shahshahani, S. et al. On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary. Commun. Math. Phys. 355, 161–243 (2017). https://doi.org/10.1007/s00220-017-2884-z

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  • DOI: https://doi.org/10.1007/s00220-017-2884-z

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