arXiv subjects

Miho Murata

Publications and source records attributed to Miho Murata.

11 recordsLinked to original sources

arXiv2026

Local and global well-posedness in the $L^2$-setting for the Q-tensor model in $\mathbb R^N$ and $\mathbb R^N_+$

The paper studies the Q-tensor model for nematic liquid crystals, a system that couples a Navier-Stokes equation with an evolution equation for the order parameter tensor Q. The first goal of the paper is to establish the local well-posedness of the system in $\mathbb R^N$ and $\mathbb R^N_+$ for $N=2,3$ in the $L^2$ framework, improving existing results in the literature, where the existence of local strong solutions was obtained only under smallness assumptions on the initial data. Fundamental is the application of the energy method, which shows a cancellation phenomena on the nonlinear terms, allowing the use of a contraction argument to prove existence and uniqueness of solutions. Finally, with the same approach we establish global well-posedness in the three-dimensional case for small initial data.

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arXiv2025

The global well-posedness for the Q-tensor model of nematic liquid crystals in the half-space

In this paper, we consider the Q-tensor model of nematic liquid crystals, which couples the Navier-Stokes equations with a parabolic-type equation describing the evolution of the directions of the anisotropic molecules, in the half-space. The aim of this paper is to prove the global well-posedness for the Q-tensor model in the $L_p$-$L_q$ framework. Our proof is based on the Banach fixed point argument. To control the higher-order terms of the solutions, we prove the weighted estimates of the solutions for the linearized problem by the maximal $L_p$-$L_q$ regularity. On the other hand, the estimates for the lower-order terms are obtained by the analytic semigroup theory. Here, the maximal $L_p$-$L_q$ regularity and the generation of an analytic semigroup are provided by the R-solvability for the resolvent problem arising from the Q-tensor model. It seems to be the first result to discuss the unique existence of a global-in-time solution for the Q-tensor model in the half-space.

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arXiv2025

The $\mathcal{R}$-boundedness of solution operators for the $Q$-tensor model of nematic liquid crystals

In this paper, we consider a resolvent problem arising from the $Q$-tensor model for liquid crystal flows in the half-space. Our purpose is to show the $\mathcal{R}$-boundedness for the solution operator families of the resolvent problem when the resolvent parameter lies near the origin. The definition of the $\mathcal{R}$-solvability implies the uniform boundedness of the operator and, consequently, the resolvent estimates for the linear system.

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arXiv2025

On the $\mathcal R$-boundedness of solution operators for a compressible fluid model of Korteweg type in general domains

In this paper, we consider a resolvent problem arising from the free boundary problem for the compressible fluid model of the Korteweg type, which is called the Navier-Stokes-Korteweg system, with surface tension in general domains. The Navier-Stokes-Korteweg system describes the liquid-vapor two-phase flow with non-zero thickness phase boundaries, which is often called the diffuse interface model. Our purpose is to show the solution operator families of the resolvent problem are $\mathcal R$-bounded, which gives us the generation of analytic semigroup and the maximal regularity in the $L_p$-in-time and $L_q$-in-space setting by applying the Weis operator valued Fourier multiplier theorem.

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arXiv2024

$\mathcal R$-bounded operator families arising from a compressible fluid model of Korteweg type with surface tension in the half-space

In this paper, we consider a resolvent problem arising from the free boundary value problem for the compressible fluid model of Korteweg type, which is called as the Navier-Stokes-Korteweg system, with surface tension in the half-space. The Navier-Stokes-Korteweg system is known as a diffuse interface model for liquid-vapor two-phase flows. Our purpose is to show the $\mathcal R$-boundedness for the solution operator families of the resolvent problem, which gives us the maximal regularity estimates in the $L_p$-in-time and $L_q$-in-space setting by applying the Weis's operator valued Fourier multiplier theorem.

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arXiv2024

The $L^p$-$L^q$ maximal regularity for the Beris-Edward model in the half-space

In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields $Q$. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal $L^p$ -$L^q$ regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the $\mathcal R$-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data.

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arXiv2022

Nondegeneracy of ground states for nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies in three and four dimensions

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies $ω$. Since the limiting profile of the ground state as $ω\to \infty$ is the Aubin-Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin-Talenti function) fails to lie in $L^{2}(\mathbb{R}^{d})$ for $d=3,4$, the nondegeneracy problem for $d=3,4$ is more difficult than that for $d\ge 5$ and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for $d=3,4$ by modifying the arguments in [2, 3]. We also show that the linearized operator around the ground state has exactly one negative eigenvalue.

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arXiv2021

Global well posedness for a Q-tensor model of nematic liquid crystals

In this paper, we prove the global well posedness and the decay estimates for a $\mathbb Q$-tensor model of nematic liquid crystals in $\mathbb R^N$, $N \geq 3$. This system is coupled system by the Navier-Stokes equations with a parabolic-type equation describing the evolution of the director fields $\mathbb Q$. The proof is based on the maximal $L_p$ -$L_q$ regularity and the $L_p$ -$L_q$ decay estimates to the linearized problem.

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arXiv2021

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter in three and four dimensions

We prove the uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent and a large frequency parameter. This study is motivated by the paper [2] and aims to remove the restriction on dimension imposed there. In this paper, we employ the fixed-point argument developed in [7] to prove the uniquness. Hence, the linearization around the Aubin-Talenti function plays a key role. Furthermore, we need some estimates for the associated perturbed resolvents (see Proposition 3.1).

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arXiv2020

The global well-posedness of the compressible fluid model of Korteweg type for the critical case

In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals to $0$ at the given constant state. It is shown that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. As a result, we also prove the decay estimates of the solutions to the nonliner problem. In order to obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for a low frequencies.

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arXiv2019

The global well-posedness for the compressible fluid model of Korteweg type

In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small initial data in $\mathbb R^N$, $N \geq 3$. In this study, the main tools are the maximal $L_p$-$L_q$ regularity and $L_p$-$L_q$ decay properties of solutions to the linearized equations.

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