### Abstract

We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝ^{n} (n ≥ 2). We report a local in time unique existence theorem in the space W^{2,1}
_{q, p} = L_{p}((0, T), W^{2}
_{q}(Ω)) ∩ W^{1}
_{q}((0, T), L_{q}(Ω)) with some T>0, 2<p<∞ and n<q<∞ for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the L_{p}-L_{q} maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179-186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223-238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137-157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672-686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207-249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227-276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222-257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189-220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307-352) and Tani (Small-time existence for the three-dimensional incompressible Navier- Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299-331).

Original language | English |
---|---|

Pages (from-to) | 201-214 |

Number of pages | 14 |

Journal | Applicable Analysis |

Volume | 90 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 Jan |

### Fingerprint

### Keywords

- Free boundary problem
- Local in time solvability
- Navier-Stokes equation
- Surface tension

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension.** / Shibata, Yoshihiro; Shimizu, Senjo.

Research output: Contribution to journal › Article

*Applicable Analysis*, vol. 90, no. 1, pp. 201-214. https://doi.org/10.1080/00036811003735899

}

TY - JOUR

T1 - Report on a local in time solvability of free surface problems for the Navier-Stokes equations with surface tension

AU - Shibata, Yoshihiro

AU - Shimizu, Senjo

PY - 2011/1

Y1 - 2011/1

N2 - We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝn (n ≥ 2). We report a local in time unique existence theorem in the space W2,1 q, p = Lp((0, T), W2 q(Ω)) ∩ W1 q((0, T), Lq(Ω)) with some T>0, 2p-Lq maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179-186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223-238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137-157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672-686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207-249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227-276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222-257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189-220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307-352) and Tani (Small-time existence for the three-dimensional incompressible Navier- Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299-331).

AB - We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝn (n ≥ 2). We report a local in time unique existence theorem in the space W2,1 q, p = Lp((0, T), W2 q(Ω)) ∩ W1 q((0, T), Lq(Ω)) with some T>0, 2p-Lq maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179-186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223-238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137-157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672-686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207-249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227-276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222-257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189-220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307-352) and Tani (Small-time existence for the three-dimensional incompressible Navier- Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299-331).

KW - Free boundary problem

KW - Local in time solvability

KW - Navier-Stokes equation

KW - Surface tension

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UR - http://www.scopus.com/inward/citedby.url?scp=78651231204&partnerID=8YFLogxK

U2 - 10.1080/00036811003735899

DO - 10.1080/00036811003735899

M3 - Article

VL - 90

SP - 201

EP - 214

JO - Applicable Analysis

JF - Applicable Analysis

SN - 0003-6811

IS - 1

ER -