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

Yoshihiro Shibata, Senjo Shimizu

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    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 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, 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 Lp-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).

    Original languageEnglish
    Pages (from-to)201-214
    Number of pages14
    JournalApplicable Analysis
    Volume90
    Issue number1
    DOIs
    Publication statusPublished - 2011 Jan

    Fingerprint

    Surface Tension
    Free Surface
    Navier Stokes equations
    Surface tension
    Solvability
    Navier-Stokes Equations
    Arch
    Arches
    Algebra
    Fluids
    Interval Algebra
    Maximal Regularity
    Fluid
    Evolution Problems
    Motion
    Compatibility Conditions
    Exterior Domain
    Incompressible Navier-Stokes Equations
    Free Boundary Problem
    Surface Waves

    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.

    In: Applicable Analysis, Vol. 90, No. 1, 01.2011, p. 201-214.

    Research output: Contribution to journalArticle

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    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).

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