Singular limits in the Cauchy problem for the damped extensible beam equation

Racke, Reinhard; Yoshikawa, Shuji

Singular limits in the Cauchy problem for the damped extensible beam equation

Dateien

Racke_0-264314.pdfGröße: 312.23 KBDownloads: 189

Datum

2014

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Working Paper/Technical Report

Publikationsstatus

Published

Zusammenfassung

We study the Cauchy problem of the Ball model for an extensible beam: [\rho \partial_t^2 u + \delta \partial_t u + \kappa \partial_x^4 u + \eta \partial_t \partial_x^4 u = \left(\alpha + \beta \int_{\R} |\partial_x u|^2 dx + \gamma \eta \int_{\R} \partial_t \partial_x u \partial_x u dx \right) \partial_x^2 u.]. The aim of this paper is to investigate singular limits as $\rho \to 0$ for this problem. In the authors' previous paper \cite{ra-yo} decay estimates of solutions $u_{\rho}$ to the equation in the case $\rho>0$ were shown. With the help of the decay estimates we describe the singular limit in the sense of the following uniform (in time) estimate: [| u_{\rho} - u_{0} |_{L^{\infty}([0,\infty); H^2(\R))} \leq C \rho.]

Fachgebiet (DDC)

510 Mathematik

Zitieren

ISO 690

RACKE, Reinhard, Shuji YOSHIKAWA, 2014. Singular limits in the Cauchy problem for the damped extensible beam equation

BibTex

@techreport{Racke2014Singu-30729,
  year={2014},
  series={Konstanzer Schriften in Mathematik},
  title={Singular limits in the Cauchy problem for the damped extensible beam equation},
  number={334},
  author={Racke, Reinhard and Yoshikawa, Shuji}
}

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    <dcterms:abstract xml:lang="eng">We study the Cauchy problem of the Ball model for an extensible beam: \[\rho \partial_t^2 u + \delta \partial_t u + \kappa \partial_x^4 u + \eta \partial_t \partial_x^4 u = \left(\alpha + \beta \int_{\R} |\partial_x u|^2 dx + \gamma \eta \int_{\R} \partial_t \partial_x u \partial_x u dx \right) \partial_x^2 u.\]. The aim of this paper is to investigate singular limits as $\rho \to 0$ for this problem. In the authors' previous paper \cite{ra-yo} decay estimates of solutions $u_{\rho}$ to the equation in the case $\rho&gt;0$ were shown. With the help of the decay estimates we describe the singular limit in the sense of the following uniform (in time) estimate: \[\| u_{\rho} - u_{0} \|_{L^{\infty}([0,\infty); H^2(\R))} \leq C \rho.\]</dcterms:abstract>
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