The linearization principle for a free boundary problem for viscous, capillary incompressible fluids.

*(English)*Zbl 1300.35077
J. Math. Sci., New York 195, No. 1, 20-60 (2013) and Zap. Nauchn. Sem. POMI 410, 36-103 (2013).

Summary: The free boundary problem associated with a viscous incompressible surface wave subject to the capillary force on a free upper surface and the Dirichlet boundary condition on a fixed bottom surface is considered. In the spatially periodic case, a general linearization principle is proved, which gives, for sufficiently small perturbations from a linearly stable stationary solution, the existence of a global solution of the associated system and the exponential convergence of the latter to the stationary one. The convergence of the velocity, the pressure, and the free boundary is proved in anisotropic Sobolev-Slobodetskii spaces, and then a suitable change of variables is performed to state the problem in a fixed domain. This linearization principle is applied to the study of the rest state’s stability in the case of general potential forces.

##### MSC:

35Q30 | Navier-Stokes equations |

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

76B45 | Capillarity (surface tension) for incompressible inviscid fluids |

35R35 | Free boundary problems for PDEs |

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\textit{S. J. N. Mosconi} and \textit{V. A. Solonnikov}, J. Math. Sci., New York 195, No. 1, 20--60 (2013; Zbl 1300.35077)

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##### References:

[1] | M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Uspekhi Mat. Nauk, 19, No. 3, 53–161 (1964). · Zbl 0137.29602 |

[2] | J. T. Beale, “The initial value problem for the Navier–Stokes equations with a free boundary,” Comm. Pure Appl. Math., 31, 359–392 (1980). · Zbl 0464.76028 |

[3] | J. T. Beale, “Large-time regularity of viscous surface waves,” Arch. Rat. Mech. Anal., 84, 307–352 (1984). · Zbl 0545.76029 · doi:10.1007/BF00250586 |

[4] | J. T. Beale and T. Nishida, “Large-time behaviour of viscous surface waves,” Lect. Notes Num. Appl. Anal., 8, 1–14 (1985). · Zbl 0642.76048 |

[5] | M. E. Bogovskii, “Resolution of some problems of the vector analysis connected with the operators div and grad,” Trudy S. L. Sobolev Semin., 1, 5–40 (1980). |

[6] | O. A. Ladyzenskaya and V. A. Solonnikov, “The linearization principle and invariant manifolds for problems in magneto-hydrodynamics,” J. Math. Sci., 8, 384–422 (1977). · Zbl 0404.35089 · doi:10.1007/BF01084611 |

[7] | T. Nishida, Y. Teramoto, and H. Yoshihara, “Global in time behaviour of viscous surface waves: horizontally periodic motion,” J. Math. Kyoto Univ., 44, 271–323 (2004). · Zbl 1095.35028 |

[8] | M. Padula and V. A. Solonnikov, “On Rayleigh–Taylor stability,” Ann. Univ. Ferrara, Sez. VIII sc. Mat., 46, 307–336 (2000). · Zbl 1001.76039 |

[9] | V. A. Solonnikov, “On the stability of uniformly rotating viscous incompressible self-gravitating liquid,” Zap. Nauchn. Semin. POMI, 348, 165–208 (2007). |

[10] | V. A. Solonnikov, “On the stability of uniformly rotating viscous incompressible self-gravitating liquid,” J. Math. Sci., 152, No. 5, 4343–4370 (2008). |

[11] | V. A. Solonnikov, “On the linear problem arising in the study of a free boundary problem for the Navier–Stokes equation,” Algebra Analiz, 22, No. 6, 235–269 (2010). |

[12] | D. Sylvester, “Large time existence of small viscous surface waves without surface tension,” Comm. Partial Diff. Eqs., 15, 823–903 (1990). · Zbl 0731.35081 · doi:10.1080/03605309908820709 |

[13] | 13.N. Tanaka and A. Tani, “Large-time existence of surface waves in incompressible viscous fluid with or without surface tension,” Arch. Rat. Mech. Anal., 130, 303–314 (1995). · Zbl 0844.76025 · doi:10.1007/BF00375142 |

[14] | A. Tani, “Small-time existence for the three-dimensional incompressible Navier–Stokes equations with a free surface,” Arch. Rat. Mech. Anal., 133, 299–331 (1996). · Zbl 0857.76026 · doi:10.1007/BF00375146 |

[15] | Y. Teramoto, “On the Navier–Stokes flow down an inclined plane,” J. Math. Kyoto Univ., 32, 593–619 (1992). · Zbl 0784.76021 |

[16] | Y. Teramoto, “The initial value problem for a viscous incompressible flow down an inclined plane,” Hiroshima Math. J., 15, 619–643 (1985). · Zbl 0625.76030 |

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