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An effective model for boundary vortices in thin-film micromagnetics (2023)
Journal Article
Kurzke, M., & Ignat, R. (2023). An effective model for boundary vortices in thin-film micromagnetics. Mathematical Models and Methods in Applied Sciences, 33(9), 1929-1973. https://doi.org/10.1142/S021820252350046X

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the mic... Read More about An effective model for boundary vortices in thin-film micromagnetics.

Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors (2023)
Journal Article
Golovaty, D., Kurzke, M., Montero, J. A., & Spirn, D. (2023). Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors. Journal of Nonlinear Science, 33(3), Article 48. https://doi.org/10.1007/s00332-023-09898-x

Tetrahedral frame fields have applications to certain classes of nematic liquid crystals and frustrated media. We consider the problem of constructing a tetrahedral frame field in three-dimensional domains in which the boundary normal vector is inclu... Read More about Tetrahedral Frame Fields via Constrained Third-Order Symmetric Tensors.

Global Jacobian and ?-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices (2021)
Journal Article
Ignat, R., & Kurzke, M. (2021). Global Jacobian and ?-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices. Journal of Functional Analysis, 280(8), Article 108928. https://doi.org/10.1016/j.jfa.2021.108928

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vort... Read More about Global Jacobian and ?-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices.

Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy (2020)
Journal Article
Ignat, R., Kurzke, M., & Lamy, X. (2020). Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy. SIAM Journal on Mathematical Analysis, 52(1), 524-542. https://doi.org/10.1137/19m1262978

For ? > 0, let u? : ? ? R 2 be a solution of the Ginzburg-Landau system ??u? = 1 ? 2 u?(1 ? |u?| 2) in a Lipschitz bounded domain ?. In an energy regime that excludes interior vortices, we prove that 1 ? |u?| is uniformly estimated by a positive powe... Read More about Global Uniform Estimate for the Modulus of Two-Dimensional Ginzburg-Landau Vortexless Solutions with Asymptotically Infinite Boundary Energy.

The effect of forest dislocations on the evolution of a phase-field model for plastic slip (2018)
Journal Article
Dondl, P., Kurzke, M., & Wojtowytsch, S. (2019). The effect of forest dislocations on the evolution of a phase-field model for plastic slip. Archive for Rational Mechanics and Analysis, 232(1), 65–119. https://doi.org/10.1007/s00205-018-1317-2

We consider the gradient flow evolution of a phase-field model for crystal dislocations in a single slip system in the presence of forest dislocations. The model is based on a Peierls-Nabarro type energy penalizing non-integer slip and elastic stress... Read More about The effect of forest dislocations on the evolution of a phase-field model for plastic slip.

Gross-Pitaevskii vortex motion with critically scaled inhomogeneities (2017)
Journal Article
Kurzke, M., Marzuola, J. L., & Spirn, D. (2017). Gross-Pitaevskii vortex motion with critically scaled inhomogeneities. SIAM Journal on Mathematical Analysis, 49(1), 471-500. https://doi.org/10.1137/15M1049014

We study the dynamics of vortices in an inhomogeneous Gross--Pitaevskii equation iδtu = Δu + 1/ε²(ρ² ε(ᵡ) - |u|²). For a unique scaling regime |ρε(ᵡ) - 1 = O(logε¯¹), it is shown that vortices can interact both with the background perturbation and wi... Read More about Gross-Pitaevskii vortex motion with critically scaled inhomogeneities.

Vortex liquids and the Ginzburg-Landau equation (2014)
Journal Article
Kurzke, M., & Spirn, D. (2014). Vortex liquids and the Ginzburg-Landau equation. Forum of Mathematics, Sigma, 2, Article e11. https://doi.org/10.1017/fms.2014.6

We establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quanti... Read More about Vortex liquids and the Ginzburg-Landau equation.