arXiv subjects

Takeshi Fukao

Publications and source records attributed to Takeshi Fukao.

27 records · Page 2Linked to original sources

arXiv2018

Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn-Hilliard systems

Condition imposed on the nonlinear terms of a nonlinear diffusion equation with {R}obin boundary condition is the main focus of this paper. The degenerate parabolic equations, such as the {S}tefan problem, the {H}ele--{S}haw problem, the porous medium equation and the fast diffusion equation, are included in this class. By characterizing this class of equations as an asymptotic limit of the {C}ahn--{H}illiard systems, the growth condition of the nonlinear term can be improved. In this paper, the existence and uniqueness of the solution are proved. From the physical view point, it is natural that, the {C}ahn--{H}illiard system is treated under the homogeneous {N}eumann boundary condition. Therefore, the {C}ahn--{H}illiard system subject to the {R}obin boundary condition looks like pointless. However, at some level of approximation, it makes sense to characterize the nonlinear diffusion equations.

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arXiv2017

Abstract approach of degenerate parabolic equations with dynamic boundary conditions

An initial boundary value problem of the nonlinear diffusion equation with a dynamic boundary condition is treated. The existence problem of the initial-boundary value problem is discussed. The main idea of the proof is an abstract approach from the evolution equation governed by the subdifferential. To apply this, the setting of suitable function spaces, more precisely the mean-zero function spaces, is important. In the case of a dynamic boundary condition, the total mass, which is the sum of volumes in the bulk and on the boundary, is a point of emphasis. The existence of a weak solution is proved on this basis.

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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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arXiv2016

Cahn-Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions

In this paper the well-posedness of some degenerate parabolic equations with a dynamic boundary condition is considered. To characterize the target degenerate parabolic equation from the Cahn-Hilliard system, the nonlinear term coming from the convex part of the double-well potential is chosen using a suitable maximal monotone graph. The main topic of this paper is the existence problem under an assumption for this maximal monotone graph for treating a wider class. The existence of a weak solution is proved.

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arXiv2015

Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems

An asymptotic limit of a class of Cahn-Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose-Fife type, fast diffusion equation and so on. Namely, by setting the suitable potential of the Cahn-Hilliard systems, all of these problems can be obtained as limits of the Cahn-Hilliard related problems. Convergence results and error estimates are proved.

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arXiv2015

Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions

This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by a limit of an equation and dynamic boundary condition of Cahn-Hilliard type. By using this Cahn-Hilliard approach, it becomes clear that the state of the mushy region of the Stefan problem is characterized by an asymptotic limit of the fourth-order system, which has a double-well structure. This fact also raises the possibility of the numerical application of the Cahn-Hilliard system to the degenerate parabolic equation, of which the Stefan problem is one.

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arXiv2015

Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution.

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arXiv2014

Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary

The well-known Cahn-Hilliard equation entails mass conservation if a suitable boundary condition is prescribed. In the case when the equation is also coupled with a dynamic boundary condition, including the Laplace-Beltrami operator on the boundary, the total mass on the inside of the domain and its trace on the boundary should be conserved. The new issue of this paper is the setting of a mass constraint on the boundary. The effect of this additional constraint is the appearance of a Lagrange multiplier; in fact, two Lagrange multipliers arise, one for the bulk, the other for the boundary. The well-posedness of the resulting Cahn-Hilliard system with dynamic boundary condition and mass constraint on the boundary is obtained. The theory of evolution equations governed by subdifferentials is exploited and a complete characterization of the solution is given.

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arXiv2014

The Allen-Cahn equation with dynamic boundary conditions and mass constraints

The Allen-Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem.

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