On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities

Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett

    Research output: Contribution to journalArticle

    18 Citations (Scopus)

    Abstract

    The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

    Original languageEnglish
    Pages (from-to)171-194
    Number of pages24
    JournalEvolution Equations and Control Theory
    Volume1
    Issue number1
    DOIs
    Publication statusPublished - 2012

    Fingerprint

    Two-phase Flow
    Incompressible Flow
    Two phase flow
    Well-posedness
    Entropy
    Phase Transition
    Phase transitions
    Semiflow
    Orbits
    Local Well-posedness
    Limit Set
    Well-defined
    Orbit
    Regularity
    Singularity
    Energy
    Model

    Keywords

    • Entropy
    • Phase transitions
    • Surface tension
    • Time weights
    • Two-phase Navier-Stokes equations
    • Well-posedness

    ASJC Scopus subject areas

    • Applied Mathematics
    • Control and Optimization
    • Modelling and Simulation

    Cite this

    On well-posedness of incompressible two-phase flows with phase transitions : The case of equal densities. / Prüss, Jan; Shibata, Yoshihiro; Shimizu, Senjo; Simonett, Gieri.

    In: Evolution Equations and Control Theory, Vol. 1, No. 1, 2012, p. 171-194.

    Research output: Contribution to journalArticle

    @article{dd3e49a36e4a458eb4e311c989d55bd5,
    title = "On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities",
    abstract = "The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.",
    keywords = "Entropy, Phase transitions, Surface tension, Time weights, Two-phase Navier-Stokes equations, Well-posedness",
    author = "Jan Pr{\"u}ss and Yoshihiro Shibata and Senjo Shimizu and Gieri Simonett",
    year = "2012",
    doi = "10.3934/eect.2012.1.171",
    language = "English",
    volume = "1",
    pages = "171--194",
    journal = "Evolution Equations and Control Theory",
    issn = "2163-2472",
    publisher = "American Institute of Mathematical Sciences",
    number = "1",

    }

    TY - JOUR

    T1 - On well-posedness of incompressible two-phase flows with phase transitions

    T2 - The case of equal densities

    AU - Prüss, Jan

    AU - Shibata, Yoshihiro

    AU - Shimizu, Senjo

    AU - Simonett, Gieri

    PY - 2012

    Y1 - 2012

    N2 - The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

    AB - The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

    KW - Entropy

    KW - Phase transitions

    KW - Surface tension

    KW - Time weights

    KW - Two-phase Navier-Stokes equations

    KW - Well-posedness

    UR - http://www.scopus.com/inward/record.url?scp=84872173936&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84872173936&partnerID=8YFLogxK

    U2 - 10.3934/eect.2012.1.171

    DO - 10.3934/eect.2012.1.171

    M3 - Article

    AN - SCOPUS:84872173936

    VL - 1

    SP - 171

    EP - 194

    JO - Evolution Equations and Control Theory

    JF - Evolution Equations and Control Theory

    SN - 2163-2472

    IS - 1

    ER -