Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

Kenji Nishihara

    研究成果: Article

    18 引用 (Scopus)

    抄録

    Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

    元の言語English
    ページ(範囲)177-196
    ページ数20
    ジャーナルRoyal Society of Edinburgh - Proceedings A
    133
    発行部数1
    出版物ステータスPublished - 2003

    Fingerprint

    Optimal Convergence Rate
    Compressible flow
    Compressible Flow
    Porous Media
    Porous materials
    Porous Medium Equation
    Convergence Rate
    Cauchy Problem
    Sufficient

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics

    これを引用

    @article{b1e8ce08dcc74e8daaeaaf9aa21833b9,
    title = "Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media",
    abstract = "Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.",
    author = "Kenji Nishihara",
    year = "2003",
    language = "English",
    volume = "133",
    pages = "177--196",
    journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",
    issn = "0308-2105",
    publisher = "Cambridge University Press",
    number = "1",

    }

    TY - JOUR

    T1 - Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

    AU - Nishihara, Kenji

    PY - 2003

    Y1 - 2003

    N2 - Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

    AB - Consider the Cauchy problem for a one-dimensional compressible flow through porous media, vt - ux = 0, x ∈ R, t > 0, ut + p(v)x = -αu, (v, u)|t=0 = (v0, u0) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v0(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥L∞ = O(t-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.

    UR - http://www.scopus.com/inward/record.url?scp=0037228601&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0037228601&partnerID=8YFLogxK

    M3 - Article

    VL - 133

    SP - 177

    EP - 196

    JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

    JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

    SN - 0308-2105

    IS - 1

    ER -