Searcharxiv⌕ Search

arXiv subjects

Yoshikazu Giga

Publications and source records attributed to Yoshikazu Giga.

At least 19 recordsLinked to original sources
arXiv2026

Solutions to a One-Dimensional Combustion-Type Free Boundary Problem via Maximal Regularity

We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.

math.AP↗
arXiv2026

On dynamic stability of energetically stable equilibria of the Navier-Stokes-Korteweg flows

We consider the Navier-Stokes-Korteweg equations in a bounded domain or a periodic cell. The pressure considered in this paper may not be monotone with respect to the density so that there exist non-constant equilibria allowing two-phases. Using a simple Hilbert space framework, we prove that if an isolated equilibrium is energetically stable and non- degenerate, it is exponentially stable under the isothermal Navier-Stokes-Korteweg flows when the space dimension is less than or equal to three. For non-isolated case, we prove that a global-in-time solution near an energetically stable equilibrium converges to possibly another equilibrium exponentially fast. No smallness assumptions on equilibria are imposed. For the proof we apply a (generalized) stability principle due to J. Prüss, M. Wilke and G. Simonett (2013).

math.AP↗
arXiv2026

On existence of a collapsed bubble with surface tension in viscous incompressible fluid

We consider the one-phase free boundary problem for the incompressible Navier-Stokes equations in $\mathbb{R}^d$ ($d\ge2$). The surface tension is taken into account. The initial domain, which is the outside of a bubble, is an exterior domain. We prove that there exists a bubble evolving by this free boundary problem which collapses in a finite time without blowing up of principal curvatures of its boundary. In other words, what is called a splash singularity is formed in a finite time. This type of result is also valid for a bounded initial domain. To construct such an example, we introduce the notion of a domain with $δ$-wing which is a flat Riemannian manifold that is not embedded in $\mathbb{R}^d$, but it covers the $δ$-neighborhood of the original domain whose boundary is self-intersected.

math.AP↗
arXiv2026

Segmentation of monotone data by Kobayashi-Warren-Carter type total variation energies

We consider a Kobayashi-Warren-Carter (KWC) type total variation energy with a fidelity term. Since the energy is non-convex, the profiles of minimizers are quite different from those of the original Rudin-Osher-Fatemi energy. In one-dimensional setting, we prove that KWC type energy (and its generalization) with fidelity must have a piecewise constant minimizer if the data in fidelity is bounded not necessarily in $BV$. Moreover, we give quantitative estimates of the energy for a monotone data in fidelity. This estimate shows that any minimizer must be piecewise constant with an improved estimate of the number of jumps for a monotone data. We also show the non-uniqueness of minimizers. Since this energy is useful from the point of segmentation or clustering, we compare with results of segmentation by the original Rudin-Osher-Fatemi energy and Mumford-Shah energy.

math.AP↗
arXiv2026

On segmentation by total variation type energies of Kobayashi-Warren-Carter type with fidelity

We consider a total variation type energy which measures the jump discontinuities different from usual total variation energy. Such a type of energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy with minimization with respect to the order parameter. We consider the Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of total variation energy. We show that all minimizers are piecewise constant if the data function in the fidelity term is continuous in one-dimensional setting. Moreover, the number of jumps is bounded by an explicit constant involving a constant related to the fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy where a minimizer has no jumps if the data has no jumps. Our results give an upper bound of the number of segments in a segmentation problem. The existence of a minimizer is guaranteed in multi-dimensional setting when the data is bounded.

math.AP↗
arXiv2026

Asymptotically self-similar graph-like solutions to a multi-dimensional surface diffusion flow equation under contact angle and no-flux boundary conditions

This paper studies Mullins' model of thermal grooving which consists of a surface diffusion flow equation with contact angle and no-flux boundary conditions. We consider this problem in a multi-dimensional half space and prove that if the slope of the initial data is close to that consistent with the contact angle, then there exists a unique global-in-time solution. In particular, we show the existence of a self-similar solution for a given behavior at the space infinity. We also show that our global solution converges to a self-similar solution as the time tends to infinity if the initial data is asymptotically homogeneous at the space infinity. No assumption on the size of the contact angle is imposed.

math.AP↗
arXiv2025

Mosco convergence framework for singular limits of gradient flows on Hilbert spaces with applications

We consider the question of convergence of a sequence of gradient flows defined on different Hilbert spaces. In order to give meaning to this idea, we introduce a notion of connecting operators. This permits us to generalize the concept of Mosco convergence of functionals to our present setting, and state a desired convergence result for gradient flows, which we then prove. We present a variety of examples, including thin domains, dynamic boundary conditions, and discrete-to-continuum limits.

math.AP↗
arXiv2025

No formulation of a new phase for a free boundary problem in combustion theory

We consider a free boundary problem for the heat equation with a given non-negative external heat source. On the free boundary, we impose the zero Dirichlet condition and the fixed normal derivative so that heat escapes from the boundary. In various settings, we show that there exist no solutions when the initial temperature equals the fixed temperature no matter where the initial location of the free boundary is given provided that the external heat source is bounded from above. We also note that there is a chance to have a solution when the external temperature is unbounded as time tends to zero by giving a self-similar solution.

math.AP↗
arXiv2025

The three limits of the hydrostatic approximation

The primitive equations are derived from the $3D$-Navier-Stokes equations by the hydrostatic approximation. Formally, assuming an $\varepsilon$-thin domain and anisotropic viscosities with vertical viscosity $ν_z=\mathcal{O}(\varepsilon^γ)$ where $γ=2$, one obtains the primitive equations with full viscosity as $\varepsilon\to 0$. Here, we take two more limit equations into consideration: For $γ<2$ the $2D$-Navier-Stokes equations are obtained. For $γ>2$ the primitive equations with only horizontal viscosity $-Δ_H$ as $\varepsilon\to 0$. Thus, there are three possible limits of the hydrostatic approximation depending on the assumption on the vertical viscosity. The latter convergence has been proven recently by Li, Titi, and Yuan using energy estimates. Here, we consider more generally $ν_z=\varepsilon^2 δ$ and show how maximal regularity methods and quadratic inequalities can be an efficient approach to the same end for $\varepsilon,δ\to 0$. The flexibility of our methods is also illustrated by the convergence for $δ\to \infty$ and $\varepsilon\to 0$ to the $2D$-Navier-Stokes equations.

math.AP↗
arXiv2025

On periodic traveling wave solutions with or without phase transition to the Navier-Stokes-Korteweg and the Euler-Korteweg equations

The Navier-Stokes-Korteweg and the Euler-Korteweg equations are considered in isothermal setting. These are diffuse interface models of two-phase flow. For the Navier-Stokes-Korteweg equations, we show that there is no periodic traveling wave solution with phase transition although there exists a non-constant periodic traveling wave solution with no phase transition. For the Euler-Korteweg equations, we show that there always exists a periodic traveling wave solution with phase transition for any period if the Korteweg relaxation parameter is small compared with the period provided that the available energy is double-well type. We also show that such a periodic traveling wave solution tends to a monotone traveling wave solution as the period tends to infinity under suitable spatial translation. Our numerical experiment suggests that there is periodic traveling wave with phase transition which is stable under periodic perturbation for small viscosity but it seems that this is a transition pattern.

math.AP↗
arXiv2025

On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data

A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ initial and boundary data for $1/p+1/p'=1$. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data.

math.AP↗
arXiv2024

Large time behavior of exponential surface diffusion flows on $\mathbb{R}$

We consider a surface diffusion flow of the form $V=\partial_s^2f(-κ)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $κ$ denotes the curvature and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$. We consider this equation for the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)κ$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $κ=0$.

math.AP↗
arXiv2024

On a dryout point for a stationary incompressible thermal fluid with phase transition in a pipe

A dryout point is recognized as the position where the phase transition from liquid to vapor occurs. In the one-dimensional case, by solving the stationary incompressible Navier-Stokes-Fourier equations with phase transition, we derive a necessary and sufficient condition for a dryout point to exist when the temperature at the liquid-vapor interface is given. In addition, we show by considering thermodynamics that the temperature at the dryout point and the density of the vapor phase can be determined by given density and sufficiently small injected mass flux of the liquid phase.

math.AP↗
arXiv2024

A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

We consider a minimizing movement scheme of Chambolle type for the mean curvature flow equation with prescribed contact angle condition in a smooth bounded domain in $\mathbb{R}^d$ ($d\geq2$). We prove that an approximate solution constructed by the proposed scheme converges to the level-set mean curvature flow with prescribed contact angle provided that the domain is convex and that the contact angle is away from zero under some control of derivatives of given prescribed angle. We actually prove that an auxiliary function corresponding to the scheme uniformly converges to a unique viscosity solution to the level-set equation with an oblique {derivative} boundary condition corresponding to the prescribed boundary condition.

math.AP↗
arXiv2024

A few topics on total variation flows

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of calibrability, finite time extinction, and some regularity properties of solutions. We focus on the second-order flow (possibly with weights) and the fourth-order flow. We also discuss the fractional cases.

math.AP↗
arXiv2023

On a diffusion equation with rupture

We propose a model to describe an evolution of a bubble cluster with rupture. In a special case, the equation is reduced to a single parabolic equation with evaporation for the thickness of a liquid layer covering bubbles. We postulate that a bubble collapses if this liquid layer becomes thin. We call this collapse a rupture. We prove for our model that there is a periodic-in-time solution if the place of rupture occurs only in the largest bubble. Numerical tests indicate that there may not exist a periodic solution if such an assumption is violated.

math.AP↗