We present a convergence result for solutions of the vector‐valued Allen‐Cahn equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multiphase mean~curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time‐integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and volume constraints.© 2018 Wiley Periodicals, Inc.

Magnetic skyrmions have been the subject of extensive experimental studies in ferromagnetic thin films and multilayers, revealing a diversity in their size, stability and internal structure. While the orthodox skyrmion theory focuses on the Dzyaloshinskii-Moryia interaction (DMI) and neglects higher-order energy terms, it is becoming clear that the full stray field energy needs to be taken into account to understand these recent observations. Here we present a micromagnetic study based on rigorous mathematical analysis which allows to account for the full stray field energy in the thin film and low DMI regime. In this regime, the skyrmion profile is close to a Belavin-Polyakov profile, which yields analytical expressions for the equilibrium skyrmion radius and energy. The obtained formulas provide a clear identification of Dzyaloshinskii-Moryia and long-range dipolar interactions as two physical mechanisms determining skyrmion size and stability, a consideration of importance for the optimization of skyrmion characteristics for spintronic applications.

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the twodimensional incompressible Navier-Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number Re and palinstrophy P which maximize dP/dt is constructed by numerically solving suitable optimization problems for a wide range of Re and P, providing numerical evidence for the sharpness of the analytic estimate dP/dt ≤ a + b √ ln Re + c P 3/2 with respect to both Re and P. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e., sup t>0 P(t), are observed to scale with the magnitude of the initial palinstrophy P(0) in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier-Stokes equation are discussed.