arXiv subjects

Shunsuke Kurima

Publications and source records attributed to Shunsuke Kurima.

At least 19 recordsLinked to original sources

arXiv2025

Analysis of a nonisothermal and conserved phase field system with inertial term

This paper deals with a conserved phase field system that couples the energy balance equation with a Cahn--Hilliard type system including temperature and the inertial term for the order parameter. In the case without inertial term, the system under study was introduced by Caginalp. The inertial term is motivated by the occurrence of rapid phase transformation processes in nonequilibrium dynamics. A double-well potential is well chosen and the related nonlinearity governing the evolution is assumed to satisfy a suitable growth condition. The viscous variant of the Cahn--Hilliard system is also considered along with the inertial term. The existence of a global solution is proved via the analysis of some approximate problems with Yosida regularizations, and the use of the Cauchy--Lipschitz--Picard theorem in an abstract setting. Moreover, we study the convergence of the system, with or without the viscous term, as the inertial coefficient tends to zero.

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arXiv2024

Nonlocal to local convergence of phase field systems with inertial term

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn--Hilliard models.

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arXiv2022

Nonlocal to local convergence of singular phase field systems of conserved type

This paper deals with a singular nonlocal phase field system of conserved type.Colli--K.\ [Nonlinear Anal.\ 190 (2020)] have derived existence of solutions to a singular phase field system of conserved type. On the other hand, Davoli--Scarpa--Trussardi [Arch. Ration. Mech. Anal.\ 239 (2021)] have studied nonlocal to local convergence of Cahn-Hilliard equations. In this paper we prove existence of solutions to a nonlocal singular phase field system of conserved type whose kernel is not $W^{1, 1}$ and focus on nonlocal to local convergence of singular phase field systems of conserved type.

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arXiv2022

Existence for a nonlocal Penrose--Fife type phase field system with inertial term

This article deals with a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove existence of solutions in reference to Colli--Grasselli--Ito [Electron. J. Differential Equations 2002, No. 100, 32 pp.] or not (see Remark 1.1). In this paper we introduce a time discretization scheme (see Section 2), pass to the limit as the time step $h$ goes to $0$ and obtain an error estimate for the difference between continuous and discrete solutions (see Section 5).

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arXiv2021

Existence for a singular nonlocal phase field system with inertial term

In this paper we deal with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $θ$ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln θ)_{t}$, the inertial term and the nonlocal diffusion term for the order parameter $φ$ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.1).

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arXiv2021

Time discretization of a nonlocal phase-field system with inertial term

Time discretizations of phase-field systems have been studied. For example, a time discretization and an error estimate for a parabolic-parabolic phase-field system have been studied by Colli--K. [Commun. Pure Appl. Anal. 18 (2019)]. Also, a time discretization and an error estimate for a simultaneous abstract evolution equation applying parabolic-hyperbolic phase field systems and the linearized equations of coupled sound and heat flow have been studied (see K. [ESAIM Math. Model. Numer. Anal.54 (2020), Electron. J. Differential Equations 2020, Paper No. 96]). On the other hand, although existence, continuous dependence estimates and behavior of solutions to nonlocal phase-field systems with inertial terms have been studied by Grasselli--Petzeltová--Schimperna [Quart. Appl. Math. 65 (2007)], time discretizations of these systems seem to be not studied yet. In this paper we focus on employing a time discretization scheme for a nonlocal phase-field system with inertial term and establishing an error estimate for the difference between continuous and discrete solutions.

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arXiv2020

Convergence of a Cahn-Hilliard type system to a parabolic-elliptic chemotaxis system with nonlinear diffusion

This paper deals with a parabolic-elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn-Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli-Visintin [Comm. Partial Differential Equations 15 (1990), 737-756] for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli-Fukao [J. Math. Anal. Appl. 429 (2015), 1190-1213]). Moreover, Colli-Fukao [J. Differential Equations 260 (2016), 6930-6959] established existence of solutions to the nonlinear diffusion equation by passing to the limit in the Cahn-Hilliard equation. However, Cahn-Hilliard approaches to chemotaxis systems with nonlinear diffusions seem not to be studied yet. This paper will try to derive existence of solutions to a parabolic-elliptic chemotaxis system with nonlinear diffusion by passing to the limit in a Cahn-Hilliard type chemotaxis system.

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arXiv2020

Time discretization of an abstract problem applying to the linearized equations of coupled sound and heat flow

In this paper we deal with an abstract problem which includes the linearized equations of coupled sound and heat flow as an example. Recently, a time discretization of a simultaneous abstract evolution equation applying to some parabolic-hyperbolic phase-field systems has been studied. This paper focuses on a time discretization of an abstract problem applying to the linearized equations of coupled sound and heat flow. Also, this paper gives some parabolic-hyperbolic phase-field systems as examples.

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arXiv2019

Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems

This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter. Although a time discretization of an initial value problem for an abstract evolution equation has been studied, time discretizations of initial value problems for simultaneous abstract evolution equations seem to be not studied yet. In this paper we focus on a time discretization of a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems. Moreover, we can establish an error estimate for the difference between continuous and discrete solutions.

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arXiv2019

Global existence for a phase separation system deduced from the entropy balance

This paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn--Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization.

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arXiv2018

Time discretization of a nonlinear phase field system in general domains

This paper deals with the nonlinear phase field system \begin{equation*} \begin{cases} \partial_t (θ+\ell φ) - Δθ= f & \mbox{in}\ Ω\times(0, T), \\[1mm] \partial_t φ- Δφ+ ξ+ π(φ) = \ell θ,\ ξ\inβ(φ) & \mbox{in}\ Ω\times(0, T) \end{cases} \end{equation*} in a general domain $Ω\subseteq\mathbb{R}^N$. Here $N \in \mathbb{N}$, $T>0$, $\ell>0$, $f$ is a source term, $β$ is a maximal monotone graph and $π$ is a Lipschitz continuous function. We note that in the above system the nonlinearity $β+π$ replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that $Ω$ is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step $h$ goes to $0$. In the limit procedure we face with the difficulty that the embedding $H^1(Ω) \hookrightarrow L^2(Ω)$ is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order $h^{1/2}$ for the difference between continuous and discrete solutions.

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arXiv2018

Asymptotic analysis for Cahn--Hilliard type phase field systems related to tumor growth in general domains

This article considers a limit system by passing to the limit in the following Cahn--Hilliard type phase field system related to tumor growth as $β\searrow0$: \begin{equation*} \begin{cases} α\partial_{t} μ_β + \partial_{t} φ_β-Δμ_β = p(σ_β - μ_β) & \mbox{in}\ Ω\times(0, T), \\[1mm] μ_β = β\partial_{t} φ_β + (-Δ+1)φ_β + ξ_β + π(φ_β),\ ξ_β \in B(φ_β) & \mbox{in}\ Ω\times(0, T), \\[1mm] \partial_{t} σ_β -Δσ_β = -p(σ_β - μ_β) & \mbox{in}\ Ω\times(0, T) \end{cases} \end{equation*} in a bounded or an unbounded domain $Ω\subset \mathbb{R}^{N}$ with smooth bounded boundary. Here $N\in\mathbb{N}$, $T>0$, $α>0$, $β>0$, $p\geq0$, $B$ is a maximal monotone graph and $π$ is a Lipschitz continuous function. In the case that $Ω$ is a bounded domain, $p$ and $-Δ+1$ are replaced with $p(φ_β)$ and $-Δ$, respectively, and $p$ is a Lipschitz continuous function, Colli--Gilardi--Rocca--Sprekels (2017) have proved existence of solutions to the limit problem with this approach by applying the Aubin--Lions lemma for the compact embedding $H^1(Ω) \hookrightarrow L^2(Ω)$ and the continuous embedding $L^2(Ω) \hookrightarrow (H^1(Ω))^{*}$. However, the Aubin--Lions lemma cannot be applied directly when $Ω$ is an unbounded domain. The present work establishes existence of weak solutions to the limit problem both in the case of bounded domains and in the case of unbounded domains. To this end we construct an applicable theory for both of these two cases by noting that the embedding $H^1(Ω) \hookrightarrow L^2(Ω)$ is not compact in the case that $Ω$ is an unbounded domain.

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arXiv2018

Existence and energy estimates of weak solutions for nonlocal Cahn--Hilliard equations on unbounded domains

This paper considers the initial-boundary value problem for the nonlocal Cahn--Hilliard equation $$ \partial_tφ+ (-Δ+1)(a(\cdot)φ-J\astφ+ G'(φ)) = 0 \quad \mbox{in}\ Ω\times(0, T) $$ in an unbounded domain $Ω\subset \mathbb{R}^N$ with smooth bounded boundary, where $N\in\mathbb{N}$, $T>0$, and $a(\cdot), J, G$ are given functions. In the case that $Ω$ is a bounded domain and $-Δ+1$ is replaced with $-Δ$, this problem has been studied by using a Faedo--Galerkin approximation scheme considering the compactness of the Neumann operator $-Δ+1$ (cf. Colli--Frigeri--Grasselli (2012), Gal--Grasselli (2014)). However, the compactness of the Neumann operator $-Δ+1$ breaks down when $Ω$ is an unbounded domain. The present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain.

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arXiv2018

Global weak solutions to a 3-dimensional degenerate and singular chemotaxis-Navier--Stokes system with logistic source

This paper considers the degenerate and singular chemotaxis-Navier--Stokes system with logistic term $n_t + u\cdot\nabla n =Δn^m - χ\nabla\cdot(n\nabla c) + κn -μn^2$, $x \in Ω,\ t>0$, $c_t + u\cdot\nabla c = Δc - nc$, $x \in Ω,\ t>0$, $u_t + (u\cdot\nabla)u = Δu + \nabla P + n\nablaΦ, \quad \nabla\cdot u = 0$, $x \in Ω,\ t>0$, where $Ω\subset \mathbb{R}^3$ is a bounded domain and $χ,κ\ge 0$ and $m, μ>0$. In the above system without fluid environment Jin (J. Differential Equations, 2017) showed existence and boundedness of global weak solutions. On the other hand, in the above system with $m=1$, Lankeit (Math.\ Models Methods Appl. Sci., 2016) established global existence of weak solutions. However, the above system with $m>0$ has not been studied yet. The purpose of this talk is to establish global existence of weak solutions in the chemotaxis-Navier--Stokes system with degenerate diffusion and logistic term.

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arXiv2017

Nonlinear diffusion equations as asymptotic limits of Cahn--Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $Ω\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-Δ+1)β(u) = g \quad \mbox{in}\ Ω\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $β$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $β$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $β$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-Δ+1)(\varepsilon(-Δ+1)u_{\varepsilon} + β(u_{\varepsilon}) + π_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ Ω\times(0, T), \] which is called the Cahn--Hilliard system, even if $Ω\subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.

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arXiv2017

Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier--Stokes system with competitive kinetics

This paper is concerned with the 3-dimensional two-species chemotaxis-Navier--Stokes system with Lotka--Volterra competitive kinetics under homogeneous Neumann boundary conditions and initial conditions. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier--Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.

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arXiv2017

Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller--Segel-Stokes system with competitive kinetics

This paper deals with the two-species Keller--Segel-Stokes system with competitive kinetics $(n_1)_t + u\cdot\nabla n_1 =Δn_1 - χ_1\nabla\cdot(n_1\nabla c)+ μ_1n_1(1- n_1 - a_1n_2)$, $(n_2)_t + u\cdot\nabla n_2 =Δn_2 - χ_2\nabla\cdot(n_2\nabla c) + μ_2n_2(1- a_2n_1 - n_2), c_t + u\cdot\nabla c =Δc - c + αn_1 +βn_2$, $u_t= Δu + \nabla P+ (γn_1 + δn_2)\nablaϕ$, $ \nabla\cdot u = 0$ under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary. Many mathematicians study chemotaxis-fluid systems and two-species chemotaxis systems with competitive kinetics. However, there are not many results on coupled two-species chemotaxis-fluid systems which have difficulties of the chemotaxis effect, the competitive kinetics and the fluid influence. Recently, in the two-species chemotaxis-Stokes system, where $-c+αn_1+βn_2$ is replaced with $-(αn_1+βn_2)c$ in the above system, global existence and asymptotic behavior of classical solutions were obtained in the 3-dimensional case under the condition that $μ_1,μ_2$ are sufficiently large. Nevertheless, the above system has not been studied yet; we cannot apply the same argument as in the previous works because of lacking the $L^\infty$-information of $c$. The main purpose of this paper is to obtain global existence and stabilization of classical solutions to the above system in the 3-dimensional case under the largeness conditions for $μ_1,μ_2$.

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arXiv2017

A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators

This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion type equations. The other is the Cahn--Hilliard type system. The present paper applies Brézis theory directly to both equations and gives existence results for these two equations even if the domain is unbounded. Moreover, an error estimate is also proved via apriori estimates obtained directly.

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