On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations

In this paper we deal with weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-Magneto-Hydrodynamic equations. First we prove that these solutions satisfy some weak-strong uniqueness property. Then we investigate the validity of energy identities. In particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation. In the case of the Hall-Magneto-Hydrodynamic equations we also give a sufficient condition to guarantee the magneto-helicity identity. Our conditions correspond to the same heuristic scaling as the one introduced by Onsager in hydrodynamic theory. Finally we examine the sign, locally, of the anomalous dissipations of weak solutions obtained by some natural approximation processes.Comment: 45 page

Adaptive mesh refinement for the Landau–Lifshitz–Gilbert equation

We propose a new adaptive algorithm for the approximation of the Landau–Lifshitz–Gilbert equation via a higher-order tangent plane scheme. We show that the adaptive approximation satisfies an energy inequality and demonstrate numerically, that the adaptive algorithm outperforms uniform approaches

Strong solutions of the Landau-Lifshitz-Bloch equation in Besov space

We focus on the existence and uniqueness of the three-dimensional Landau-Lifshitz-Bloch equation supplemented with the initial data in Besov space $\dot{B}_{2,1}^{\frac{3}{2}}$. Utilizing a new commutator estimate, we establish the local existence and uniqueness of strong solutions for any initial data in $\dot{B}_{2,1}^{\frac{3}{2}}$. When the initial data is small enough in $\dot{B}_{2,1}^{\frac{3}{2}}$, we obtain the global existence and uniqueness. Furthermore, we also establish a blow-up criterion of the solution to the Landau-Lifshitz-Bloch equation and then we prove the global existence of strong solutions in Sobolev space under a new condition based on the blow-up criterion.Comment: 20 page

FEM-BEM coupling for Maxwell–Landau–Lifshitz–Gilbert equations via convolution quadrature: Weak form and numerical approximation

The Maxwell equations in the unbounded three dimensional space are coupled to the Landau-Lifshitz-Gilbert equation on a (not necessarily convex) bounded domain. A weak formulation of the whole coupled system is derived based on the boundary integral formulation of the exterior Maxwell equations. We show existence of a weak solution and uniqueness of the Maxwell part of the weak solution. A numerical algorithm is proposed based on finite elements and boundary elements as spatial discretisation and using the backward Euler method and convolution quadratures for the interior domain and the boundary, respectively. Well-posedness and convergence of the numerical algorithm are shown, under minimal assumptions on the regularity of solutions. Numerical experiments illustrate and expand on the theoretical results

FEM-BEM coupling for Maxwell–Landau–Lifshitz–Gilbert equations via convolution quadrature: Weak form and numerical approximation

The Maxwell equations in the unbounded three dimensional space are coupled to the Landau-Lifshitz-Gilbert equation on a (not necessarily convex) bounded domain. A weak formulation of the whole coupled system is derived based on the boundary integral formulation of the exterior Maxwell equations. We show existence of a weak solution and uniqueness of the Maxwell part of the weak solution. A numerical algorithm is proposed based on finite elements and boundary elements as spatial discretisation and using the backward Euler method and convolution quadratures for the interior domain and the boundary, respectively. Well-posedness and convergence of the numerical algorithm are shown, under minimal assumptions on the regularity of solutions. Numerical experiments illustrate and expand on the theoretical results

Relaxed optimal control for the stochastic Landau-Lifshitz-Gilbert equation

We consider the stochastic Landau-Lifshitz-Gilbert equation, perturbed by a real-valued Wiener process. We add an external control to the effective field as an attempt to drive the magnetization to a desired state and also to control thermal fluctuations. We use the theory of Young measures to relax the given control problem along with the associated cost. We consider a control operator that can depend (possibly non-linearly) on both the control and the associated solution. Moreover, we consider a fairly general associated cost functional without any special convexity assumption. We use certain compactness arguments, along with the Jakubowski version of the Skorohod Theorem to show that the relaxed problem admits an optimal control

Variational Methods for Evolution

The meeting focused on the last advances in the applications of variational methods to evolution problems governed by partial differential equations. The talks covered a broad range of topics, including large deviation and variational principles, rate-independent evolutions and gradient flows, heat flows in metric-measure spaces, propagation of fracture, applications of optimal transport and entropy-entropy dissipation methods, phase-transitions, viscous approximation, and singular-perturbation problems