arXiv subjects

Yoshihiro Tonegawa

Publications and source records attributed to Yoshihiro Tonegawa.

22 records · Page 2Linked to original sources

arXiv2012

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.

math.AP↗

arXiv2012

A second derivative Hölder estimate for weak mean curvature flow

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show C^{2,α} regularity almost everywhere.

math.AP↗

arXiv2011

Two-phase flow problem coupled with mean curvature flow

We prove the existence of generalized solution for incompressible and viscous non-Newtonian two-phase fluid flow for spatial dimension 2 and 3. The phase boundary moves along with the fluid flow plus its mean curvature while exerting surface tension force to the fluid. An approximation scheme combining the Galerkin method and the phase field method is adopted.

math.AP↗

arXiv2010

Stable phase interfaces in the van der Waals--Cahn--Hilliard theory

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and an embedded smooth stable minimal hypersurface away from a closed singular set of co-dimension at least 7 in general dimensions. This result was previously known in case the critical points are local minimizers of energy, in which case the limit-hypersurface is locally area minimizing and its (normalized) multiplicity is 1 a.e. Our theorem uses earlier work of the first author establishing stability of the limit-interface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.

math.DG↗