Searcharxiv⌕ Search

arXiv subjects

Yoshihiro Shibata

Publications and source records attributed to Yoshihiro Shibata.

At least 19 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.

math.AP↗
arXiv2025

Free Boundary Problem for inhomogeneous Navier-Stokes equations

We study free boundary problems for incompressible inhomogeneous flows governed by the Navier--Stokes equations, focusing on the regularity and global-in-time well-posedness of solutions in critical functional frameworks for small initial data. We introduce a novel analytical framework for free boundary problems formulated as perturbations of the half-space. Our approach relies on the natural Lagrangian change of coordinates and a detailed analysis of the linearized problem (the Stokes system) in the maximal regularity regime, formulated in the Lebesgue spaces $L_p(0,T; L_q)$, including time-weighted variants. The main difficulty lies in the treatment of boundary terms, for which we apply a new technique based on complex interpolation to control nonlinear terms in fractional Sobolev spaces. This strategy also allows us to handle the case of variable density, which is not easily addressed by approaches based on Besov spaces. Using this framework and real interpolation techniques, we construct also solutions in the Lorentz class $L_{p,1}(0,T; L_q)$ in time. The method further enables a rigorous study of the stability of equilibrium configurations. In particular, we resolve the problem in two spatial dimensions, where the interplay between geometry and regularity is especially subtle. Beyond these specific applications, the proposed approach provides a powerful tool for broader classes of nonlinear PDEs and further developments in maximal regularity theory.

math.AP↗
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.

math.AP↗
arXiv2025

Maximal $L_1$-regularity of the Navier-Stokes equations with free boundary conditions via a generalized semigroup theory

This paper develops a new approach to show the maximal regularity theorem of the Stokes equations with free boundary conditions in the half-space $\mathbb R^d_+$, $d \ge 2$, within the $L_1$-in-time and $\mathcal B^s_{q, 1}$-in-space framework with $(q, s)$ satisfying $1 < q < \infty$ and $1 + 1 / q < s < 1 / q$, where $\mathcal B^s_{q, 1}$ stands for either homogeneous or inhomogeneous Besov spaces. In particular, we establish a generalized semigroup theory within an $L_1$-in-time and $\mathcal B^s_{q,1}$-in-space framework, which extends a classical $C_0$-analytic semigroup theory to the case of inhomogeneous boundary conditions. The maximal $L_1$-regularity theorem is proved by estimating the Fourier--Laplace inverse transform of the solution to the generalized Stokes resolvent problem with inhomogeneous boundary conditions, where density and interpolation arguments are used. The maximal $L_1$-regularity theorem is applied to show the unique existence of a local strong solution to the Navier--Stokes equations with free boundary conditions for arbitrary initial data $\boldsymbol a$ in $B^s_{q, 1} (\mathbb R^d_+)^d$, where $q$ and $s$ satisfy $d-1 < q \le d$ and $-1+d/q < s < 1/q$, respectively. If we assume that the initial data $\boldsymbol a$ are small in $\dot B^{1 + d / q}_{q, 1} (\mathbb R^d_+)^d$, $d 1 < q < 2 d$, then the unique existence of a global strong solution to the system is proved.

math.AP↗
arXiv2024

$L_1$ approach to the compressible viscous fluid flows in general domains

We prove the $L_1$ in time and $B^{s+1}_{q,1}\times B^s_{q,1}$ in space maximal regularity for the Stokes equations in the viscous compressible fluid flows in domains in the $N$ dimensional Euclidean space $R^N$ whose boundary is $C^3$ compact hypersurface. As an application, the local well-posedness is proved for the compressible Navier-Stokes equations with Dirichlet condition. Danchin and Tolksdorf have studied the same problem in a bounded domains. Since they used Da-Prato and Grisvard theory directly, they need the assumption that the domain is compact. We imvestigate a new method to obtain $L_1$ maximal regularity which is based on real interpolation theories and thanks to this method, we can remove the compactness assumptions in the study due to Danchin and Torksdorf. Our method can be applied to obtain $L_1$ in time maximal regularity theorem for the initial boundary value problem of the system of parabolic or hyperbolic-parabolic equations with non-homogeneous boundary conditions.

math.AP↗
arXiv2023

$L_1$ approach to the compressible viscous fluid flows in the half-space

In this paper, we proved the local well-posedness for the Navier-Stokes equtions describing the motion of isotropic barotoropic compressible viscous fluid flow with non-slip boundary conditions, wehre the fluid domain is the half-space in the $N$-dimensional Euclidean space. The density part of solutions and their time derivative belong to $L_1$ in time with some Besov spaces in space and also the velosity parts and their time derivative belong to $L_1$ in time with some Besov spaces in space. We use Lagrange transformation to eliminate the covection term and we use an analytic semgroup approach. Our Stokes semigroup is not only a continuous analytic semigroup but also has an $L_1$ in times maximal regularity with some Besov spaces in space.

math.AP↗
arXiv2023

$L_1$ approach to the compressible viscous fluid flows in the half-space

In this paper, we prove the local well-posedness for the Navier-Stokes equations describing the motion of isotropic barotoropic compressible viscous fluid flow with non-slip boundary conditions, where the fluid domain is the $N$ dimensional half-sapce. We solve the equations in the $L_1$ in time and Besov spaces $B^s_{q,1}$ in space maximal regularity framework. Here, we assume that $-1+N/q \leq s < 1/q$ and $N-1 < q < 2N$. We use Lagrange transformation to eliminate the convection term and we use an analytic semigroup approach. We only assume the strictly positiveness of initial mass density.

math.AP↗
arXiv2023

On the global wellposedness of free boundary problem for the Navier-Stokes system with surface tension

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time $t$ tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain $Ω_t\subset\mathbf{R}^3$, $t>0$, to a problem in the lower half-space $\mathbf{R}_-^3$. We then establish some time-weighted estimate of solutions, in an $L_p$-in-time and $L_q$-in-space setting, for the linearized problem around the trivial steady state with the help of $L_r\text{-}L_s$ time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in $\mathbf{R}_-^3$ admits a global-in-time solution in the $L_p\text{-}L_q$ setting and that the solution decays polynomially as time $t$ tends to infinity under the assumption that $p$, $q$ satisfy the conditions: $2<p<\infty$, $3<q<16/5$, and $(2/p)+(3/q)<1$. Finally, we apply the inverse transformation of Hanzawa's one to the solution in $\mathbf{R}_-^3$ to prove our main results mentioned above for the original problem in $Ω_t$. Here we want to emphasize that it is not allowed to take $p=q$ in the above assumption about $p$, $q$, which means that the different exponents $p$, $q$ of $L_p\text{-}L_q$ setting play an essential role in our approach.

math.AP↗
arXiv2023

Global solvability for viscous free surface flows of infinite depth in three and higher dimensions

This paper is concerned with the global solvability for the Navier-Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. We first prove time weighted estimates of solutions to a linearized system of the Navier-Stokes equations by time decay estimates of a $C_0$-analytic semigroup and maximal regularity estimates in an $L_p$-in-time and $L_q$-in-space setting with suitable $p$, $q$. The time weighted estimates then enable us to show the global solvability of the Navier-Stokes equations for small initial data by the contraction mapping principle.

math.AP↗
arXiv2023

Viscous flow past a translating body with oscillating boundary

We study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable R-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions.

math.AP↗
arXiv2022

Periodic Lp estimates by R-boundedness: Applications to the Navier-Stokes equations

General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic $L^p$ estimates of maximal regularity type are established from $\mathscr{R}$-bounds of the family of solution operators ($\mathscr{R}$-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier-Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.

math.AP↗
arXiv2022

On the global well-posedness and decay of a free boundary problem of the Navier-Stokes equation in unbounded domains

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier-Stokes equations in $L_p$ in time and $L_q$ in space framework in a uniformly $H^2_\infty$ domain $Ω\subset \BR^N$ for $N\geq4$. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the $L_q$-$L_r$ estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata \cite{Shiba17CIME}. The restriction $N\geq 4$ is required to deduce an estimate for the nonlinear term $\bG(\bu)$ arising from $\divv\bv=0$. However, we establish the results in the half space $\hsp$ for $N\geq 3$ by reducing the linearized problem to the problem with $\bG=\fb0$, where $\bG$ is the right member corresponding to $\bG(\bu)$.

math.AP↗
arXiv2022

Global wellposedness of the 3D compressible Navier-Stokes equations with free surface in the maximal regularity class

This paper concerns the global well posedness issue of the Navier-Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal $L_p$-$L_q$ estimate and the $L_p$-$L_q$ decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal $L_p$-$L_q$ regularity class for some $p>2$ and $q>3.$ Namely, the solution is bounded as $L_p$ in time and $L_q$ in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal $L_p$-$L_q$ regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.

math.AP↗
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.

math.AP↗
arXiv2021

New thought on Matsumura-Nishida theory in the $L_p$-$L_q$ maximalregularity framework

In this paper, we prove the global wellposedness of the Navier-Stokes equations describing a motion of compressible, viscous, barotropic fluid flow in a 3 dim. exterior domain in the $L_p$ in time and $L_2 \cap L_6$ maximal regularity framework. This is an extension of a famous thoerem due to Matsumura-Nishida Commun Math. Phys. 89 (1983), 445--464. In Matsumura and Nishida theory, they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to $L_2$ in space-time. On the other hand, in the present manuscript space derivatives of the mass density up to first order and the space derivatives of the velocity fields up to second order belong to $L_2$ in maximal and $L_2 \cap L_6$ in space. The proof is based on the $L_p$-$L_q$ maximal regularity and decay properties of solutions to the linearized equations, namely Stokes equations appering in the study of compressible fluid flows.

math.AP↗
arXiv2021

Local Well-posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. The free boundary problem for MHD is an important problem not only for mathematical fluid dynamics but also some application to the field of engineering In fact, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. In this paper, an electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in the maximal regularity class, and in particular, the regularity class of velocity fields is one more higher than the regularity class of magnetic fields. To prove our main result, we use Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions.

math.AP↗
arXiv2021

The $L_p$-$L_q$ decay estimate for the multidimensional compressible flow with free surface in the exterior domain

The aim of this paper is to develop the general $L_p$ theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in $\mathbb{R}^N$ ($N\geq 3$). By the spectral analysis, we obtain the classical $L_p$-$L_q$ decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.

math.AP↗