https://neckpinch.com/authors/kiichi-tashiro/New arXiv papers by Kiichi Tashiro, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2608.05760arxiv:2608.05760Thu, 06 Aug 2026 00:00:00 GMTKiichi Tashiro — In this note, we show that a weak mean curvature flow with critical forcing term obtained by Liu–Tonegawa (2024) satisfies the BV-type area change formula, that is, their flow is not only a Brakke flow but also a generalized BV flow, which is proposed by Stuvard–Tonegawa (2024). To establish this result, we identify minimal conditions under which a Brakke flow satisfies the area-change formula. As an application of our main theorem, we derive a lower bound for the extinction time of generalized BV flows with a critical forcing term. We also outline the proof of a compactness theorem for generalized BV flows.Mean curvature flowWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/2512.14437arxiv:2512.14437Tue, 16 Dec 2025 00:00:00 GMTJingeon An-Lacroix, Kiichi Tashiro — It is known that there is a strong relation between the parabolic Allen–Cahn equation and the mean curvature flow, in the sense that the parabolic Allen–Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow v=−H-∂_νlog |∇u|+f(u)/|∇u|, satisfied by level surfaces of any solution to the nonlinear parabolic equation ∂ₜu=Δu-f(u). Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen–Cahn equation \{beginalignedat2 ∂ₜu&=Δu &&in \|u|<1\ |∇u|&=1/ε &&on ∂\|u|<1\, endalignedat and confirm that under reasonable assumptions, the C^α norm of the forcing term ∂_νlog|∇u| converges to zero at an algebraic rate as ε→ 0, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in ε and in time) in the C^(2,α) sense.Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryPhase fields & multiphase flows https://arxiv.org/abs/2410.12283arxiv:2410.12283Wed, 16 Oct 2024 00:00:00 GMTKiichi Tashiro — In the present paper, we study the existence of Brakke-type weak mean curvature flow satisfying a prescribed contact angle condition for a general angle θ∈ ( 0 , π) via Ilmanen's regularization. The main ingredients of the result are the extension of Ilmanen's regularization to the capillarity and the derivation of the first variation estimates for the interior and wetted boundary varifolds separately.Mean curvature flowWeak & singular flowsFree boundary & capillary https://arxiv.org/abs/2305.12374arxiv:2305.12374Sun, 21 May 2023 00:00:00 GMTKiichi Tashiro — We investigate a mean curvature flow obtained via elliptic regularization, and prove that it is not only a Brakke flow, but additionally a generalized BV flow proposed by Stuvard and Tonegawa. In particular, we show that the change in volume of the evolving phase can be expressed in terms of the generalized mean curvature of the Brakke flow.Mean curvature flowWeak & singular flows