##
**On the motion of the free surface of a liquid.**
*(English)*
Zbl 1031.35116

This is a very solid and deep paper dealing with the free boundary problem of an incompressible perfect fluid. The velocity \(v\) of the fluid obeys the Euler equations inside the moving domain and equals the normal velocity of the boundary on the free boundary. The directional derivative of the pressure \(p\) along the exterior unit normal to the boundary is assumed to be negative, namely \(\nabla_{\mathcal{N}}p \leq -\varepsilon<0\). This is a natural physical condition since the pressure has to be positive in the interior of the fluid. The major purpose of this paper is to establish global a priori bounds in Sobolev spaces for this free boundary problem supplemented with an initial datum for \(v\). It is worth pointing out that the results of this paper hold for the case when the vorticity of the fluid is not zero.

To prove their results, the authors introduce a sequence of energies \(\{E_r\}\), each of which consists of an interior part and a boundary part. The advantage of such energies is to avoid the use of fractional Sobolev spaces. The authors then show that the time derivative of \(E_r\) is bounded by a constant \(C\) times \(\sum_{s=0}^r E_s\), where \(C\) depends on \(1/\varepsilon\), and the geometry and regularity of the free boundary. A crucial fact in obtaining this bound is that the time derivative of the interior part, after integration by parts, cancels the leading order term in the time derivative of the boundary part. The global a priori bounds for \(E_r\) are then proved iteratively. In the course of the proof, significant efforts are devoted to controlling the geometry and regularity of the boundary. For this purpose, the authors work in both Eulerian coordinates and the Lagrangian coordinates.

To prove their results, the authors introduce a sequence of energies \(\{E_r\}\), each of which consists of an interior part and a boundary part. The advantage of such energies is to avoid the use of fractional Sobolev spaces. The authors then show that the time derivative of \(E_r\) is bounded by a constant \(C\) times \(\sum_{s=0}^r E_s\), where \(C\) depends on \(1/\varepsilon\), and the geometry and regularity of the free boundary. A crucial fact in obtaining this bound is that the time derivative of the interior part, after integration by parts, cancels the leading order term in the time derivative of the boundary part. The global a priori bounds for \(E_r\) are then proved iteratively. In the course of the proof, significant efforts are devoted to controlling the geometry and regularity of the boundary. For this purpose, the authors work in both Eulerian coordinates and the Lagrangian coordinates.

Reviewer: Jiahong Wu (Stillwater)

### MSC:

35Q35 | PDEs in connection with fluid mechanics |

76U05 | General theory of rotating fluids |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

PDF
BibTeX
XML
Cite

\textit{D. Christodoulou} and \textit{H. Lindblad}, Commun. Pure Appl. Math. 53, No. 12, 1536--1602 (2000; Zbl 1031.35116)

Full Text:
DOI

### References:

[1] | Baker, J Fluid Mech 252 pp 51– (1993) |

[2] | Baouendi, Comm Partial Differential Equations 2 pp 1151– (1977) |

[3] | Beale, Comm Pure Appl Math 46 pp 1269– (1993) |

[4] | Christodoulou, Arch Rational Mech Anal 130 pp 343– (1995) |

[5] | ; The global nonlinear stability of the Minkowski space. Princeton Mathematical Series, 41. Princeton University Press, Princeton, N.J., 1993. |

[6] | ; The free boundary problem for an irrotational incompressible fluid. Preprint, May 1996. |

[7] | Craig, Comm Partial Differential Equations 10 pp 787– (1985) |

[8] | Ebin, Comm Partial Differential Equations 12 pp 1175– (1987) |

[9] | Oral communication. 1997. |

[10] | Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, R.I., 1998. |

[11] | ; Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin–New York, 1983. · Zbl 0361.35003 |

[12] | The Cauchy-Poisson problem. Dinamika Splošn Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami (1974), 104-210, 254. |

[13] | Nishida, J Differential Geom 12 pp 629– (1977) |

[14] | ; Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, Mass., 1994. · Zbl 0830.53001 |

[15] | Sulem, Comm Math Phys 80 pp 485– (1981) |

[16] | Wu, Invent Math 130 pp 39– (1997) |

[17] | Wu, J Amer Math Soc 12 pp 445– (1999) |

[18] | Yosihara, Publ Res Inst Math Sci 18 pp 49– (1982) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.