This note corrects the statement and the proof of point (1) in Theorem 1.1 of [ Interfaces Free Bound . 25 (2023), no. 3, 455–490].
In this paper, we aim to provide a rigorous mathematical proof of the Weissenberg effect. We consider a liquid filling the semi-infinite annular region between two concentric cylinders rotating at different steady speeds. The behavior of the fluid leads to different shapes of the free surface. For a Newtonian fluid, the meniscus descends due to centrifugal forces. However, for a certain non-Newtonian fluid, the meniscus climbs up the inner cylinder phenomenon known as the Weissenberg effect. In a previous paper (2022), the author studied the case of a Newtonian fluid and developed a rigorous mathematical theory. In the present paper, we focus on the non-Newtonian case. By decomposing the problem into a system consisting of the Stokes problem and a transport equation, we prove the convergence of the formal perturbation series introduced by Joseph and Fosdick (1973), thereby providing a rigorous justification of the Weissenberg effect.
We propose an anisotropic diffuse interface model for three-dimensional grain boundary motion in polycrystals. The model is derived as the L-2-gradient flow of an R x S-3 constrained free energy functional. We include the case of neither smooth nor strictly convex anisotropies, thus allowing for crystalline ones. Our mathematical analysis of the corresponding system of PDEs yields existence of solutions in any fixed time interval as the first main result of the paper. In the second part, we obtain two different uniqueness results, for a particular case of anisotropies, based on some regularity of the solutions to the system. In the case that the domain is one-dimensional, we obtain uniqueness as a consequence of Lipschitz regularity. In the general N-dimensional case, we prove local in time uniqueness (and existence) of classical solutions for smooth and compatible initial data.
We study the evolution of hypersurfaces whose rate of deformation in the inward normal direction is proportional to the scalar curvature at each point. We will present the C-infinity-regularity and preservation of strict convexity for hypersurfaces evolving under the scalar curvature flow before collapse. We also have C-1,C-1-optimal regularity for convex hypersurfaces with a flat spot and local upper and lower curvature bounds in the strictly convex region.
We study the discrete-to-continuum limit of a frustrated ferromagnetic/anti-ferromagnetic S- 2 -valued spin system on the lattice lambda(n)Z(2) as lambda(n)-> 0. For S- 2 spin systems near the Landau- Lifschitz point (where the helimagnetic/ferromagnetic transition occurs) it is well known that chirality transitions emerge with vanishing energy. Inspired by recent advances on the N-clock model, we consider a spin system in which the spins are constrained to k(n) copies of S- 1 covering S( 2 )as n -> infinity. We identify a critical energy-scaling regime and a threshold on the divergence rate of k(n) -> +infinity, below which the Gamma-limit of the discrete energies captures chirality transitions while preserving an S- 2 -valued energy description in the continuum limit. To achieve this, we establish a connection with the variational analysis of a discrete approximation of a vector-valued Modica-Mortola-type functional, where the k(n) disconnected wells converge in the Hausdorff sense to a connected set as n -> 1.
