Landau-Lifshitz-Gilbert-Vlasov-Maxwell system

Dorešić, Tvrtko; Melcher, Christof (Thesis advisor); Frank, Martin (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022


The miniaturization process of transistors which are used in almost all modern electronic devices is currently approaching an inevitable breaking point. The reached scale of only a few nanometers is facing physical limitations due to quantum tunneling effects resulting in standby current leakage. The area of spin electronics has attracted a lot of interest over the past few decades as a promising new direction for further development of information storage and information processing technologies. Special attention is given to magnetic topological solitons which are envisaged to be potential information carriers manipulated using ultralow current densities. We examine the current driven magnetization dynamics in the framework of chiral and frustrated magnets which are both known to host magnetic topological solitons such as skyrmions and hopfions. Chiral magnets include the Heisenberg exchange interaction and the stabilizing Dzyaloshinski-Moriya interaction while frustrated magnets have competing first and second order gradient interactions. Recent models suggest the gyro-coupling of the Landau-Lifshitz-Gilbert equation describing magnetization and the classical electron transport governed by the Vlasov-Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields which add up to the conventional Maxwell fields. We construct a local in time unique solution to the coupled system in high order Sobolev space for compactly supported initial electron distributions. In view of Sobolev embedding, this smooth solution preserves the topology of the initial magnetization field.We study the natural energy law of the coupled system to explore global well posedness questions. Combining uniform bounds implied by energy dissipation with tools from transport theory like velocity averaging and the method of renormalization, we construct a topology conserving global weak solution in frustrated magnets. The key analytical feature of the frustrated magnet is the $H^2$ coercivity of its underlying micromagnetic interaction energy. The question of uniqueness remains an open problem for weak solutions of the Vlasov-Maxwell system alone. However, we obtain a partial uniqueness result, i.e. uniqueness of weak solutions to the LLG equation for a fixed current density.In the setting of chiral magnets, the natural energy bound does not provide enough regularity to obtain a global weak solution. We examine the weakest regularity criterion for the current density still implying existence of a weak solution.


  • Department of Mathematics [110000]
  • Chair of Applied Analysis [113110]