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 language | English |
|---|---|
| Pages (from-to) | 1-23 |
| Number of pages | 23 |
| Journal | Journal of Differential Equations |
| Volume | 243 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2007 Dec 1 |
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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
Cite this
On an Lp approach to the stationary and nonstationary problems of the Ginzburg-Landau-Maxwell equations. / Akiyama, Takahiro; Shibata, Yoshihiro.
In: Journal of Differential Equations, Vol. 243, No. 1, 01.12.2007, p. 1-23.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On an Lp approach to the stationary and nonstationary problems of the Ginzburg-Landau-Maxwell equations
AU - Akiyama, Takahiro
AU - Shibata, Yoshihiro
PY - 2007/12/1
Y1 - 2007/12/1
N2 - 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.
AB - 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.
KW - Coulomb gauge
KW - Ginzburg-Landau-Maxwell equations
KW - Global existence
KW - Integral equations
KW - L-L estimate
KW - Nonstationary problem
KW - Perfect conducting wall condition
KW - Stationary problem
KW - Superconductivity
UR - http://www.scopus.com/inward/record.url?scp=35548976992&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=35548976992&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2005.09.005
DO - 10.1016/j.jde.2005.09.005
M3 - Article
VL - 243
SP - 1
EP - 23
JO - Journal of Differential Equations
JF - Journal of Differential Equations
SN - 0022-0396
IS - 1
ER -