arXiv subjects

Ken Furukawa

Publications and source records attributed to Ken Furukawa.

11 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.

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arXiv2025

Well-Posedness of the Cauchy Problem for One-Dimensional Nonlinear Diffusion Equations with Dynamic and Fourth-Type Boundary Conditions in the Lp Lq Maximal Regularity Setting

This paper addresses the local well-posedness of the Cauchy problem for a one-dimensional diffusion equation equipped with a dynamic boundary condition and an additional boundary condition that renders the one-dimensional Laplace operator self-adjoint. The equation serves as a model for describing filtration in aquaria, originally introduced by the author and Kitahata. The boundary condition treated in this work differs from classical types such as Dirichlet, Neumann, and Robin conditions; we refer to it as the fourth or FK-type boundary condition. The boundary condition is designed to capture interactions between the two boundaries in the context of filtration. The framework for establishing well-posedness is based on L^p-L^q maximal regularity classes.

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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.

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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.

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arXiv2024

Data Assimilation to the Primitive Equations in $H^2$

In this paper we prove that the solution to the primitive equations is predicted by the corresponding data assimilation(DA) equations in $H^2$. Although, the DA equation does not include the direct information about the base solution and its initial conditions, the solution to the DA equation exponentially convergence to the base(original) solution when the external forces are known even before they are observed. Additionally, when the external force is not completely known but its spatially dense observations are available, then the DA is stable, $i.e.$ the DA solution lies in a sufficiently small neighborhood of the base solution.

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arXiv2024

Modeling and Mathematical Analysis of the Clogging Phenomenon in Filtration Filters Installed in Aquaria

This paper proposes a mathematical model for replicating a simple dynamics in an aquarium with two components; bacteria and organic matter. The model is based on a system of partial differential equations (PDEs) with four components: the drift-diffusion equation, the dynamic boundary condition, the fourth boundary condition, and the prey-predator model. The system of PDEs is structured to represent typical dynamics, including the increase of organic matter in the aquarium due to the excretion of organisms ($e.g$. fish), its adsorption into the filtration filter, and the decomposition action of the organic matter both on the filtration filter and within the aquarium. In this paper, we prove the well-posedness of the system and show some results of numerical experiments. The numerical experiments provide a validity of the modeling and demonstrate filter clogging phenomena. We compare the feeding rate with the filtration performance of the filter. The model exhibits convergence to a bounded steady state when the feed rate is reasonable, and grow up to an unbounded solution when the feeding is excessively high. The latter corresponds to the clogging phenomenon of the filter.

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arXiv2022

Data Assimilation to the Primitive Equations with $L^p$-$L^q$-based Maximal Regularity Approach

In this paper, we show mathematical justification of the data assimilation of nudging type in $L^p$-$L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations by data assimilation converges to the true solution with exponential order on the Besov space $B^{2/q}_{q,p}(Ω)$ in the periodic layer domain $Ω= \mathbb{T}^2 \times (-h, 0)$.

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arXiv2021

Justification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^\infty_H L^q_{x_3}(\Torus^3)$

The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic approximation. In this paper, we give a justification of the hydrostatic approximation by the scaled Navier-Stoke equations in anisotropic spaces $L^\infty_H L^q_{x_3} (\Torus^3)$ for $q \geq 1$.

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arXiv2020

The Hydrostatic Approximation for the Primitive Equations by the Scaled Navier-Stokes Equations under the No-Slip Boundary Condition

In this paper we justify the hydrostatic approximation of the primitive equations in the maximal $L^p$-$L^q$-setting in the three-dimensional layer domain $Ω= \Torus^2 \times (-1, 1)$ under the no-slip (Dirichlet) boundary condition in any time interval $(0, T)$ for $T>0$. We show that the solution to the scaled Navier-Stokes equations with Besov initial data $u_0 \in B^{s}_{q,p}(Ω)$ for $s > 2 - 2/p + 1/ q$ converges to the solution to the primitive equations with the same initial data in $\mathbb{E}_1 (T) = W^{1, p}(0, T ; L^q (Ω)) \cap L^p(0, T ; W^{2, q} (Ω)) $ with order $O(ε)$ where $(p,q) \in (1,\infty)^2$ satisfies $ \frac{1}{p} \leq \min \bracket{ 1 - 1/q, 3/2 - 2/q }$. The global well-posedness of the scaled Navier-Stokes equations in $\mathbb{E}_1 (T)$ is also proved for sufficiently small $ε>0$. Note that $T = \infty$ is included.

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arXiv2020

Maximal $L_p$-$L_q$ regularity for the Quasi-Steady Elliptic Problems

In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal $L_p$-$L_q$ regularity for these problems and give examples that our results are applicable. The Lopatinskii--Shapiro and the asymptotic Lopatinskii--Shapiro conditions are important to get boundedness of solution operators.

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arXiv2018

Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier-Stokes equations

Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height $\varepsilon $ with initial data $u_0=(v_0,w_0)\in B^{2-2/p}_{q,p}$, $1/q+1/p\le 1$ if $q\ge 2$ and $4/3q+2/3p\le 1$ if $q\le 2$, converges as $\varepsilon \to 0$ with convergence rate $\mathcal{O} (\varepsilon )$ to the horizontal velocity of the solution to the primitive equations with initial data $v_0$ with respect to the maximal-$L^p$-$L^q$-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the $L^2$-$L^2$-setting. The approach presented here does not rely on second order energy estimates but on maximal $L^p$-$L^q$-estimates for the heat equation.

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