arXiv subjects

Naoto Kajiwara

Publications and source records attributed to Naoto Kajiwara.

10 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).

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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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arXiv2022

Maximal $L_p$-$L_q$ regularity for the Stokes equations with various boundary conditions in the half space

We prove resolvent $L_p$ estimates and maximal $L_p$-$L_q$ regularity estimates for the Stokes equations with Dirichlet, Neumann and Robin boundary conditions in the half space. Each solution is constructed by a Fourier multiplier of $x'$-direction and an integral of $x_N$-direction. We decompose the solution such that the symbols of the Fourier multipliers are bounded and holomorphic. We see that the operator norms are dominated by a homogeneous function of order $-1$ for $x_N$-direction. The basis are Weis's operator-valued Fourier multiplier theorem and a boundedness of a kernel operator. We give a new simple approach to get maximal regularity in the half space.

math.AP↗

arXiv2022

Solution formula for generalized two-phase Stokes equations and its applications to maximal regularity; model problems

In this paper we give a solution formula for the two-phase Stokes equations with and without surface tension and gravity in the whole space with flat interface. The solution formula has already considered by Shibata-Shimizu. However we reconstruct the formula so that we are able to prove resolvent estimate and maximal regularity estimate. In the previous work, they needed to assume additional conditions on normal components. We also take care of normal components, while the assumption becomes weaker than before. The method is based on an $H^\infty$ calculus which has already used for the Stokes problems with various boundary conditions in the half space.

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arXiv2020

Higher regularity for parabolic equations based on maximal L_p-L_q spaces

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.

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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.

math.AP↗

arXiv2018

Strong time-periodic solutions to the bidomain equations with arbitrary large forces

We prove the existence of strong time-periodic solutions to the bidomain equations with arbitrary large forces. We construct weak time-periodic solutions by a Galerkin method combined with Brouwer's fixed point theorem and a priori estimate independent of approximation. We then show their regularity so that our solution is a strong time-periodic solution in $L^2$ spaces. Our strategy is based on the weak-strong uniqueness method.

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arXiv2017

Strong Time Periodic Solutions to the Bidomain Equations with FitzHugh-Nagumo Type Nonlinearities

Consider the bidomain equations subject to ionic transport described by the models of FitzHugh-Nagumo, Aliev-Panfilov, or Rogers-McCulloch. It is proved that this set of equations admits a unique, strong T-periodic solution provided it is innervated by T-periodic intra- and extracellular currents. The approach relies on a new periodic version of the classical Da Prato-Grisvard theorem on maximal L^p-regularity in real interpolation spaces.

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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.

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