arXiv subjects

Yoshikazu Giga

Publications and source records attributed to Yoshikazu Giga.

63 records · Page 4Linked to original sources

arXiv2016

On a resolvent estimate for bidomain operators and its applications

We study bidomain equations that are commonly used as a model to represent the electrophysiological wave propagation in the heart. We prove existence, uniqueness and regularity of a strong solution in $L^p$ spaces. For this purpose we derive an $L^\infty$ resolvent estimate for the bidomain operator by using a contradiction argument based on a blow-up argument. Interpolating with the standard $L^2$-theory, we conclude that bidomain operators generate $C_0$-analytic semigroups in $L^p$ spaces, which leads to construct a strong solution to a bidomain equation in $L^p$ spaces.

math.AP↗

arXiv2016

A level set crystalline mean curvature flow of surfaces

We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.

math.AP↗

arXiv2015

On asymptotic speed of solutions to level-set mean curvature flow equations with driving and source terms

We investigate a model equation in the crystal growth, which is described by a level-set mean curvature flow equation with driving and source terms. We establish the well-posedness of solutions, and study the asymptotic speed. Interestingly, a new type of nonlinear phenomena in terms of asymptotic speed of solutions appears, which is very sensitive to the shapes of source terms.

math.AP↗

arXiv2014

Stokes Resolvent Estimates in Spaces of Bounded Functions

The Stokes equation on a domain $Ω\subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1<p<\infty$. The situation is very different for the case $p=\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. Nevertheless it was recently proved by the first and the second author of this paper by a contradiction argument that the Stokes operator generates an analytic semigroup on spaces of bounded functions for a large class of domains. This paper presents a new approach as well as new a priori $L^\infty$-type estimates to the Stokes equation. They imply in particular that the Stokes operator generates a $C_0$-analytic semigroup of angle $π/2$ on $C_{0,σ}(Ω)$, or a non-$C_0$-analytic semigroup on $L^\infty_σ(Ω)$ for a large class of domains. The approach presented is inspired by the so called Masuda-Stewart technique for elliptic operators. It is shown furthermore that the method presented applies also to different type of boundary conditions as, e.g., Robin boundary conditions.

math.AP↗

arXiv2013

Lorentz space estimates for vector fields with divergence and curl in Hardy spaces

In this note, we establish the estimate on the Lorentz space $L(3/2,1)$ for vector fields in bounded domains under the assumption that the normal or the tangential component of the vector fields on the boundary vanishing. We prove that the $L(3/2,1)$ norm of the vector field can be controlled by the norms of its divergence and curl in the atomic Hardy spaces and the $L^1$ norm of the vector field itself.

math.FA↗

arXiv2013

A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion

We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed. We study the vorticity equations instead of the original Navier-Stokes equations. As an application, we extend the geometric regularity criterion for the Navier-Stokes equations in the three-dimensional half space under the no-slip boundary condition.

math.AP↗

arXiv2013

Anisotropic total variation flow of non-divergence type on a higher dimensional torus

We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with diffusion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.

math.AP↗

arXiv2013

On representation of boundary integrals involving the mean curvature for mean-convex domains

Given a mean-convex domain $Ω\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial Ω} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean curvature of $\partial Ω$ and $f$ is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.

math.AP↗

arXiv2013

Periodic total variation flow of non-divergence type in Rn

We introduce a new notion of viscosity solutions for a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of arbitrary dimension, whose diffusion on flat parts with zero slope is so strong that it becomes a nonlocal quantity. The problems include the classical total variation flow and a motion of a surface by a crystalline mean curvature. We establish a comparison principle, the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data.

math.AP↗