Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.Comment: 24 page

Low-Dose Intravenous Alteplase in Wake-Up Stroke

Background and Purpose—We assessed whether lower-dose alteplase at 0.6 mg/kg is efficacious and safe for acute fluid-attenuated inversion recovery-negative stroke with unknown time of onset. Methods—This was an investigator-initiated, multicenter, randomized, open-label, blinded-end point trial. Patients met the standard indication criteria for intravenous thrombolysis other than a time last-known-well >4.5 hours (eg, wake-up stroke). Patients were randomly assigned (1:1) to receive alteplase at 0.6 mg/kg or standard medical treatment if magnetic resonance imaging showed acute ischemic lesion on diffusion-weighted imaging and no marked corresponding hyperintensity on fluid-attenuated inversion recovery. The primary outcome was a favorable outcome (90-day modified Rankin Scale score of 0–1). Results—Following the early stop and positive results of the WAKE-UP trial (Efficacy and Safety of MRI-Based Thrombolysis in Wake-Up Stroke), this trial was prematurely terminated with 131 of the anticipated 300 patients (55 women; mean age, 74.4±12.2 years). Favorable outcome was comparable between the alteplase group (32/68, 47.1%) and the control group (28/58, 48.3%; relative risk [RR], 0.97 [95% CI, 0.68–1.41]; P=0.892). Symptomatic intracranial hemorrhage within 22 to 36 hours occurred in 1/71 and 0/60 (RR, infinity [95% CI, 0.06 to infinity]; P>0.999), respectively. Death at 90 days occurred in 2/71 and 2/60 (RR, 0.85 [95% CI, 0.06–12.58]; P>0.999), respectively. Conclusions—No difference in favorable outcome was seen between alteplase and control groups among patients with ischemic stroke with unknown time of onset. The safety of alteplase at 0.6 mg/kg was comparable to that of standard treatment. Early study termination precludes any definitive conclusions

A FINER SINGULAR LIMIT OF A SINGLE-WELL MODICA-MORTOLA FUNCTIONAL AND ITS APPLICATIONS TO THE KOBAYASHI-WARREN-CARTER ENERGY

An explicit representation of the Gamma limit of a single-well Modica-Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L1-convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness is useful to characterize the limit of minimizers the Kobayashi-Warren-Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problem is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper

On a singular limit of the Kobayashi–Warren–Carter energy

By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi–Warren–Carter energy is given in a multi-dimensional domain. A key step is to study the Gamma limit of a single-well Modica–Mortola functional. The convergence introduced here is called the sliced graph convergence, which is finer than conventional L1 convergence, and the problem is reduced to a one-dimensional setting by a slicing argument

A new numerical scheme for constrained total variation flows and its convergence

In this paper, we propose a new numerical scheme for a spatially discrete model of constrained total variation flows, which are total variation flows whose values are constrained in a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex and it is hard to calculate the minimizer of the functional with the manifold constraint. We overcome this difficulty by “localization technique" using the exponential map and prove the finite-time error estimate in general situation. Finally, we show a few numerical results for the cases that the target manifolds are S2 and SO(3)

On a singular limit of the Kobayashi--Warren--Carter energy

By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi-Warren-Carter energy is given in a multi-dimensional domain. A key step is to study the Gamma limit of a single-well Modica-Mortola functional. The convergence introduced here is called the sliced graph convergence, which is finer than conventional $L^1$ convergence, and the problem is reduced to a one-dimensional setting by a slicing argument.Comment: 31 pages, 3 figure