Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

Tej-eddine Ghoul, S. Ibrahim, V. T. Nguyen

    Research output: Contribution to journalArticle

    Abstract

    We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

    Original languageEnglish (US)
    Pages (from-to)2968-3047
    Number of pages80
    JournalJournal of Differential Equations
    Volume265
    Issue number7
    DOIs
    StatePublished - Oct 5 2018

    Fingerprint

    Blow-up Solution
    Wave equations
    Nonlinear equations
    Modulation
    Energy
    Brouwer Fixed Point Theorem
    Semilinear Wave Equation
    Energy Method
    Stationary Solutions
    Nonlinear Equations
    Symmetry

    Keywords

    • Blowup profile
    • Blowup solution
    • Stability
    • Wave maps

    ASJC Scopus subject areas

    • Analysis

    Cite this

    Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps. / Ghoul, Tej-eddine; Ibrahim, S.; Nguyen, V. T.

    In: Journal of Differential Equations, Vol. 265, No. 7, 05.10.2018, p. 2968-3047.

    Research output: Contribution to journalArticle

    Ghoul, Tej-eddine ; Ibrahim, S. ; Nguyen, V. T. / Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps. In: Journal of Differential Equations. 2018 ; Vol. 265, No. 7. pp. 2968-3047.
    @article{c688e07ee77e46349a6947472b64be83,
    title = "Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps",
    abstract = "We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha{\"e}l and Rodnianski [49] for the energy supercritical nonlinear Schr{\"o}dinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.",
    keywords = "Blowup profile, Blowup solution, Stability, Wave maps",
    author = "Tej-eddine Ghoul and S. Ibrahim and Nguyen, {V. T.}",
    year = "2018",
    month = "10",
    day = "5",
    doi = "10.1016/j.jde.2018.04.058",
    language = "English (US)",
    volume = "265",
    pages = "2968--3047",
    journal = "Journal of Differential Equations",
    issn = "0022-0396",
    publisher = "Academic Press Inc.",
    number = "7",

    }

    TY - JOUR

    T1 - Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

    AU - Ghoul, Tej-eddine

    AU - Ibrahim, S.

    AU - Nguyen, V. T.

    PY - 2018/10/5

    Y1 - 2018/10/5

    N2 - We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

    AB - We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[Formula presented]sin⁡(2u). We construct for this equation a family of C∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)∼Q([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼cu(T−t)[Formula presented],ℓ∈N⁎,ℓ>γ=γ(d)∈(1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

    KW - Blowup profile

    KW - Blowup solution

    KW - Stability

    KW - Wave maps

    UR - http://www.scopus.com/inward/record.url?scp=85047061519&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=85047061519&partnerID=8YFLogxK

    U2 - 10.1016/j.jde.2018.04.058

    DO - 10.1016/j.jde.2018.04.058

    M3 - Article

    VL - 265

    SP - 2968

    EP - 3047

    JO - Journal of Differential Equations

    JF - Journal of Differential Equations

    SN - 0022-0396

    IS - 7

    ER -