Landau-Lifshitz-Gilbert EquationCopyright: © Lehrstuhl f. Angewandte Analysis
The Landau-Lifshitz-Gilbert equation is the fundamental equation of evolution in ferromagnetism.
Mathematically, it is a hybride of the heat and Schrödinger flow of harmonic maps to the sphere. A main feature of harmonic flows is the occurrence of singularities in finite time. The interplay between dissipative and Hamiltonian dynamics leads to interesting dynamical structures, but also to analytical challenges.
Our contributions in this area include the first general existence and partial regularity result in 3D with optimal dimensionality of singular sets and the first global well-posedness result in critical Sobolev spaces. Another line of research concerns patterns and waves and dimensionality reduction (thin film limits) leading to dissipative wave mapping equations.
Our current focus is on the Landau-Lifshitz dynamics of topological solitons and the effects of spin-transfer torques in the deterministic and stochastic cases. Our current projects within the newly established CRC are dedicated to the interplay of analysis and numerics in the study of singular solutions.
- Global weak solutions for the Landau–Lifshitz–Gilbert–Vlasov–Maxwell system coupled via emergent electromagnetic fields
T Dorešić, C Melcher
Journal of Evolution Equations 22 (3), 1-32 (2022)
- Global dissipative half-harmonic flows into spheres: small data in critical Sobolev spaces
C Melcher, ZN Sakellaris
Communications in Partial Differential Equations 44 (5), 397-415 (2019)
- Strong solvability of regularized stochastic Landau–Lifshitz–Gilbert equation
O Chugreeva, C Melcher
IMA Journal of Applied Mathematics 83 (2), 261-282 (2018)
- Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equation in higher dimensions
Indiana University Mathematics Journal, 1175-1200 (2012)
- Wave-type dynamics in ferromagnetic thin films and the motion of Néel walls
A Capella, C Melcher, F Otto
Nonlinearity 20 (11), 2519 (2007)
- Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3
Communications in Partial Differential Equations 30 (4), 567-587 (2005)