On an Lp approach to the stationary and nonstationary problems of the Ginzburg-Landau-Maxwell equations

Takahiro Akiyama, Yoshihiro Shibata

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

In this paper, we study a stationary and a nonstationary problem of the Ginzburg-Landau-Maxwell equations with Coulomb gauge in the Lp framework. First we prove a unique existence of stationary solution near the constant state with a small external magnetic field. Moreover, we prove a globally in time existence of solutions to the time dependent Ginzburg-Landau-Maxwell equations with small initial data and external magnetic field, and we show its convergence to the corresponding stationary solution when time tends to infinity. The key of our approach is to use various Lp-Lq estimates of the analytic semigroup generated by the linearized problem. Especially our initial data belong to L3 without any additional regularity.

Original languageEnglish
Pages (from-to)1-23
Number of pages23
JournalJournal of Differential Equations
Volume243
Issue number1
DOIs
Publication statusPublished - 2007 Dec 1

Keywords

  • Coulomb gauge
  • Ginzburg-Landau-Maxwell equations
  • Global existence
  • Integral equations
  • L-L estimate
  • Nonstationary problem
  • Perfect conducting wall condition
  • Stationary problem
  • Superconductivity

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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