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
Under a Creative Commons license

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 and 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 and Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[3]	Kush Kinra, Manil T. Mohan. Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021061
[4]	Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations and Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297
[5]	Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345
[6]	Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355
[7]	Thomas Y. Hou, Pingwen Zhang. Convergence of a boundary integral method for 3-D water waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 1-34. doi: 10.3934/dcdsb.2002.2.1
[8]	Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149
[9]	Thomas März, Andreas Weinmann. Model-based reconstruction for magnetic particle imaging in 2D and 3D. Inverse Problems and Imaging, 2016, 10 (4) : 1087-1110. doi: 10.3934/ipi.2016033
[10]	Juan Manuel Reyes, Alberto Ruiz. Reconstruction of the singularities of a potential from backscattering data in 2D and 3D. Inverse Problems and Imaging, 2012, 6 (2) : 321-355. doi: 10.3934/ipi.2012.6.321
[11]	A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769
[12]	Gerhard Kirsten. Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations. Journal of Computational Dynamics, 2022, 9 (2) : 159-183. doi: 10.3934/jcd.2021025
[13]	Thomas Y. Hou, Danping Yang, Hongyu Ran. Multiscale analysis in Lagrangian formulation for the 2-D incompressible Euler equation. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1153-1186. doi: 10.3934/dcds.2005.13.1153
[14]	Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure and Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[15]	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
[16]	Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001
[17]	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 and Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
[18]	Claude Bardos, E. S. Titi. Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 185-197. doi: 10.3934/dcdss.2010.3.185
[19]	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 and Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127
[20]	Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

Impact Factor:

Tools

Metrics

Other articles
by authors

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

[Back to Top]