Lipschitz continuous solutions of some doubly nonlinear parabolic equations

Mitsuharu Otani, Yoshie Sugiyama

    Research output: Contribution to journalArticle

    9 Citations (Scopus)

    Abstract

    This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W0 1,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L-energy method", which provides a main tool for our arguments and would be useful for other situations.

    Original languageEnglish
    Pages (from-to)647-670
    Number of pages24
    JournalDiscrete and Continuous Dynamical Systems
    Volume8
    Issue number3
    Publication statusPublished - 2002

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    Continuous Solution
    Nonlinear Parabolic Equations
    Lipschitz
    Porous materials
    Porous Medium Equation
    Local Solution
    Non-Newtonian Fluid
    Energy Method
    P-Laplacian
    Mean Curvature
    Classical Solution
    Degeneracy
    Global Solution
    Filtration
    Evolution Equation
    Weak Solution
    Fluids
    Non-negative
    Singularity
    Term

    Keywords

    • Degenerate
    • Doubly nonlinear
    • Gradient estimate
    • Parabolic

    ASJC Scopus subject areas

    • Mathematics(all)
    • Analysis
    • Applied Mathematics
    • Discrete Mathematics and Combinatorics

    Cite this

    Lipschitz continuous solutions of some doubly nonlinear parabolic equations. / Otani, Mitsuharu; Sugiyama, Yoshie.

    In: Discrete and Continuous Dynamical Systems, Vol. 8, No. 3, 2002, p. 647-670.

    Research output: Contribution to journalArticle

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    AB - This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W0 1,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.

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