3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients

Previous Article
Anisotropic Gevrey regularity for mKdV on the circle
PROC Home
This Issue
Next Article
New regularizing approach to determining the influence coefficient matrix for gas-turbine engines

2011, 2011(Special): 624-633. doi: 10.3934/proc.2011.2011.624

3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients

Rejeb Hadiji ^1, and Ken Shirakawa ^2,

1.	Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées, CNRS UMR 8050, UFR des Sciences et Technologie, 61, Avenue du Général de Gaulle, P3, 4e étage, 94010 Créteil Cedex
2.	Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan

Received July 2010 Revised February 2011 Published October 2011

Abstract
Related Papers

In this paper, a class of minimization problems, labeled by an index 0 < $h$ < 1, is considered. Each minimization problem is for a free-energy, motivated by the magnetics in 3D-ferromagnetic thin film, and in the context, the index $h$ denotes the thickness of the observing film. The Main Theorem consists of two themes, which are concerned with the study of the solvability (existence of minimizers) and the 3D-2D asymptotic analysis for our minimization problems. These themes will be discussed under degenerate setting of the material coecients, and such degenerate situation makes the energy-domain be variable with respect to $h$. In conclusion, assuming some restrictive conditions for the domain-variation, a denite association between our 3D-minimization problems, for very thin $h$, and a 2D-limiting problem, as $h \searrow$ 0, will be demonstrated with help from the theory of $\Gamma$-convergence.

Keywords: 3D-2D asymptotic analysis, degenerate free-energy in micromagnetics, $\Gamma$-convergence.

Mathematics Subject Classification: Primary: 74G65, 35J70; Secondary: 74K35, 82D40.

Citation: Rejeb Hadiji, Ken Shirakawa. 3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients. Conference Publications, 2011, 2011 (Special) : 624-633. doi: 10.3934/proc.2011.2011.624

[1]	Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401
[2]	Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[3]	Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
[4]	Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345
[5]	Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
[6]	Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1
[7]	Thomas März, Andreas Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems & Imaging, 2016, 10 (4) : 1087-1110. doi: 10.3934/ipi.2016033
[8]	Juan Manuel Reyes, Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D. Inverse Problems & Imaging, 2012, 6 (2) : 321-355. doi: 10.3934/ipi.2012.6.321
[9]	A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769
[10]	Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153
[11]	Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[12]	Yue Cao. Blow-up criterion for the 3D viscous polytropic fluids with degenerate viscosities. Electronic Research Archive, 2020, 28 (1) : 27-46. doi: 10.3934/era.2020003
[13]	Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
[14]	Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[15]	Claude Bardos, E. S. Titi. Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 185-197. doi: 10.3934/dcdss.2010.3.185
[16]	Aseel Farhat, M. S Jolly, Evelyn Lunasin. Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127
[17]	Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
[18]	Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[19]	Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[20]	Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Rejeb Hadiji Ken Shirakawa
on Google Scholar
Rejeb Hadiji Ken Shirakawa

[Back to Top]