The weak solutions to the evolution problems of harmonic maps

Chen, Yunmei

doi:10.1007/bf01161995

Cited by 108 publications

(111 citation statements)

References 3 publications

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“…that can be obtained applying the previous regularity (7) to the equations (1) and (2). The existence of (u ε , d ε ) can be proved ( [5]) by means of three main arguments: a semi-galerkin method (space-discretization of the u-system (2), remaining the d-system (1) in the continuous sense), a maximum principle for the d-system in order to obtain the constraint |d ε | ≤ 1 and the following energy inequality,…”

Section: ε-Approximate Solutions and Dissipativitymentioning

confidence: 99%

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Global solution of nematic liquid crystals models

Guillén-González

Rojas-Medar

2002

Comptes Rendus. Mathématique

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Section: ε-Approximate Solutions and Dissipativitymentioning

confidence: 99%

“…In this Note, we establish the well-posedness in the large of a nematic liquid crystals model (formulated for instance in [7]) by means of a penalisation argument using a simplified Ericksen-Leslie model with the Ginzburg-Landau approximation [1,2].…”

Section: Introductionmentioning

confidence: 99%

Global solution of nematic liquid crystals models

Guillén-González

Rojas-Medar

2002

Comptes Rendus. Mathématique

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“…Choose now a single λ < λ 0 , and let C λ be the corresponding value of the polynomial. This gives a quantitative version of (6), namely, (11) (recall that the sequence is defined by m (n+1) = m λ , if m = m (n) ). At the same time, since the energy is bounded from below, the series n>0 w (n) 2…”

Section: Remarkmentioning

confidence: 99%

“…A magnetization distribution that satisfies (4) will be called hereafter a critical point of the energy. As for harmonic maps into spheres (see for instance [11]), critical points with finite energy equivalently satisfy the following equation in the sense of distributions…”

Section: Model Algorithm and Convergencementioning

confidence: 99%

Energetics and switching of quasi-uniform states in small ferromagnetic particles

Alouges¹,

Conti²,

DeSimone³

et al. 2004

ESAIM: M2AN

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Abstract. We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.Mathematics Subject Classification. 65L60, 78M10, 82D40.

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“…The definition of a weak solution, particularly the energy inequality (vi), is motivated by the notion of weak solutions for the harmonic map heat flow problem discussed in [11,40].…”

Section: Remark 21 (Harmonic Maps)mentioning

confidence: 99%

A finite element scheme for the evolution of orientational order in fluid membranes

2009

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Abstract.We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as well as the coupling of the local order of the constituent molecules of the membrane to its curvature. We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II 7 (1997) 1509-1520; N. Uchida, Phys. Rev. E 66 (2002) 040902] which replaces a Ginzburg-Landau penalization for the length of the order parameter by a rigid constraint. We introduce a fully discrete scheme, consisting of piecewise linear finite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence (up to subsequences) thereby proving the existence of a weak solution to the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.Mathematics Subject Classification. 35K55, 74K15.

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The weak solutions to the evolution problems of harmonic maps

Cited by 108 publications

References 3 publications

Global solution of nematic liquid crystals models

Global solution of nematic liquid crystals models

Energetics and switching of quasi-uniform states in small ferromagnetic particles

A finite element scheme for the evolution of orientational order in fluid membranes

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