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Giovanni Di Fratta

Publications and source records attributed to Giovanni Di Fratta.

33 records · Page 2Linked to original sources
arXiv2020

Spin-diffusion model for micromagnetics in the limit of long times

In this paper, we consider spin-diffusion Landau-Lifshitz-Gilbert equations (SDLLG), which consist of the time-dependent Landau-Lifshitz-Gilbert (LLG) equation coupled with a time-dependent diffusion equation for the electron spin accumulation. The model takes into account the diffusion process of the spin accumulation in the magnetization dynamics of ferromagnetic multilayers. We prove that in the limit of long times, the system reduces to simpler equations in which the LLG equation is coupled to a nonlinear and nonlocal steady-state equation, referred to as SLLG. As a by-product, the existence of global weak solutions to the SLLG equation is obtained. Moreover, we prove weak-strong uniqueness of solutions of SLLG, i.e., all weak solutions coincide with the (unique) strong solution as long as the latter exists in time. The results provide a solid mathematical ground to the qualitative behavior originally predicted by Zhang, Levy, and Fert in [Physical Review Letters 88 (2002)] in ferromagnetic multilayers.

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arXiv2020

Weak-strong uniqueness for the Landau-Lifshitz-Gilbert equation in micromagnetics

We consider the time-dependent Landau-Lifshitz-Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii-Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak-strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.

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arXiv2019

Parking 3-sphere swimmer II. The long arm asymptotic regime

The paper carries on our previous investigations on the complementary version of Purcell's rotator: a low-Reynolds-number swimmer composed of three balls of equal radii. In the asymptotic regime of very long arms, the Stokes induced governing dynamics is derived and then experimented in the context of energy minimizing self-propulsion characterized in the first part of the paper.

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arXiv2019

Variational principles of micromagnetics revisited

We revisit the basic variational formulation of the minimization problem associated with the micromagnetic energy, with an emphasis on the treatment of the stray field contribution to the energy, which is intrinsically non-local. Under minimal assumptions, we establish three distinct variational principles for the stray field energy: a minimax principle involving magnetic scalar potential and two minimization principles involving magnetic vector potential. We then apply our formulations to the dimension reduction problem for thin ferromagnetic shells of arbitrary shapes.

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arXiv2019

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

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arXiv2019

Landau-de Gennes corrections to the Oseen-Frank theory of nematic liquid crystals

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of $Γ$-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing $Q$-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy.

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arXiv2017

Linear second-order IMEX-type integrator for the (eddy current) Landau-Lifshitz-Gilbert equation

Combining ideas from [Alouges et al. (Numer. Math., 128, 2014)] and [Praetorius et al. (Comput. Math. Appl., 2017)], we propose a numerical algorithm for the integration of the nonlinear and time-dependent Landau-Lifshitz-Gilbert (LLG) equation which is unconditionally convergent, formally (almost) second-order in time, and requires only the solution of one linear system per time-step. Only the exchange contribution is integrated implicitly in time, while the lower-order contributions like the computationally expensive stray field are treated explicitly in time. Then, we extend the scheme to the coupled system of the Landau-Lifshitz-Gilbert equation with the eddy current approximation of Maxwell equations (ELLG). Unlike existing schemes for this system, the new integrator is unconditionally convergent, (almost) second-order in time, and requires only the solution of two linear systems per time-step.

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arXiv2017

Modular inequalities for the maximal operator in variable Lebesgue spaces

A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the Hardy-Littlewood maximal function in $L^{p(\cdot)}(\mathbb{R}^n)$ holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality \[ \int_ΩMf(x)^{p(x)}\,dx \ \leq c_1 \int_Ω|f(x)|^{q(x)}\,dx + c_2, \] where $c_1,\,c_2$ are non-negative constants and $Ω$ is any measurable subset of $\mathbb{R}^n$. As a corollary we get sufficient conditions for the modular inequality \[ \int_Ω|Tf(x)|^{p(x)}\,dx \ \leq c_1 \int_Ω|f(x)|^{q(x)}\,dx + c_2, \] where $T$ is any operator that is bounded on $L^p(Ω)$, $1<p<\infty$.

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arXiv2017

Dimension reduction for the micromagnetic energy functional on curved thin films

Recently a significant interest in ferromagnetic curved thin films has appeared. In particular, thin spherical shells are currently of great interest due to their capability to support skyrmion solutions which can be stabilized by curvature effects only, in contrast to the planar case where the Dzyaloshinskii-Moriya interaction is required. This paper aims to a $Γ$-asymptotic analysis of the micromagnetic energy functional, when the shell is generated, like in the case of a sphere, by a bounded, convex and smooth surface.

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arXiv2016

Parking 3-sphere swimmer. I. Energy minimizing strokes

The paper is about the parking 3-sphere swimmer ($\text{sPr}_3$). This is a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of $\text{sPr}_3$ with angles of $120^{\circ}$ . The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e. closed curves of the form $t\in [0,2π] \mapsto (\cos t)u + (\sin t)v$ for suitable orthogonal vectors $u$ and $v$ of $\mathbb{R}^3$. A simple analytic expression for the vectors $u$ and $v$ is derived. The results of the paper are used in a second article where the real physical dynamics of $\text{sPr}_3$ is analyzed in the asymptotic range of very long arms.

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arXiv2016

Cell averaging two-scale convergence: Applications to periodic homogenization

The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest: bounded sequences in $L^2_{\sharp}[Y,L^2(Ω)]$ and $L^2_{\sharp}[Y,H^1(Ω)]$ are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for $\mathbb{S}^2$-valued stationary harmonic maps.

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arXiv2016

The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid

The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of $\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its physical relevance, a separate section is devoted to the the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism.

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arXiv2014

Homogenization of Composite Ferromagnetic Materials

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents. These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications. The main objective of this paper is to perform, by means of Gamma-convergence and two-scale convergence, a rigorous derivation of the homogenized Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. We thus describe the Gamma-limit of the Gibbs-Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

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