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Hideo Kozono

Publications and source records attributed to Hideo Kozono.

11 recordsLinked to original sources
arXiv2026

Asymptotic behavior of solutions to elliptic equations in 2D exterior domains

The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $L^{p,q}$ or the weak Lebesgue space $L^{p,\infty}$ with certain conditions on the coefficients, we give natural and an almost sharp pointwise estimate of the solution at spacial infinity. The proof is based on the argument by Korobkov--Pileckas--Russo [4], in which the decay property of the solution to the vorticity equation of the two-dimensional Navier--Stokes equations was studied.

math.AP↗
arXiv2026

Incompressible Euler equations in 3D bounded domains in a critical space

We consider the 3D incompressible Euler equations in bounded domains $Ω$ with smooth boundary $\partialΩ$. Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space $B^s_{p, q}(A)$ by means of the Stokes operator $A$ with the Neumann boundary condition on $\partialΩ$, and prove unique local existence theorem of strong solution for the initial data in the critical Besov space $B^{\frac52}_{2, 1}(A)$. Our proof relies on the method of vanishing viscosity. The commutator estimate plays an essential role for derivation of energy bounds which hold uniformly with respect to viscosity constants.

math.AP↗
arXiv2026

Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains

Based on the analysis by Iwabuchi-Matsuyama-Taniguchi (2019), we first introduce our framework of Besov spaces $\dot B^s_{p, q}$ on the bounded domain $Ω\subset {\mathbb R}^d$ with smooth boundary $\partial Ω$ in terms of the Stokes operator $A=A_2$ with the Neumann boundary condition on $\partialΩ$ in $L^2_σ(Ω)$. Under some geometric assumption on $Ω$, we establish $L^p-L^q$ type estimates of the semi-group $\{e^{-tA}\}_{t \ge 0}$ in $\dot B^s_{p, q}$ and prove a local well-posedness of the Navier-Stokes equations with the initial data in $\dot B^{-1+\frac dp }_{p, q}$ for $d < p < \infty$ and $1 \le q \le \infty$. Since $d < p$, we have $L^{d, \infty} \subset \dot B^{-1+\frac dp }_{p, \infty}$ so that our space for well-posedness is larger than any other previous one in bounded domains.

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arXiv2025

Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.

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arXiv2023

Liouville-type theorems for the new Taylor--Couette flow of the stationary Navier--Stokes equations

We study the stationary Navier--Stokes equations in the region between two rotating concentric cylinders. We first prove that, under the small Reynolds number, if the fluid is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily a generalized Taylor-Couette flow which is a new exact solution of the Navier--Stokes equations. If, in addition, the associated pressure is bounded or periodic in the $z$-axis, then it coincides with the well-known canonical Taylor-Couette flow. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such as the fluid is indeed, necessarily axisymmetric. It is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the new exact form of the Taylor-Couette flow.

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arXiv2022

Asymptotic behavior and Liouville-type theorems for axisymmetric stationary Navier-Stokes equations outside of an infinite cylinder with a periodic boundary condition

We study the asymptotic behavior of solutions to the steady Navier-Stokes equations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that the flow is periodic in $x_3$-direction and has no swirl. This problem is closely related with two-dimensional exterior problem. Under a condition on the generalized finite Dirichlet integral, we give a pointwise decay estimate of the vorticity at the spatial infinity. Moreover, we prove a Liouville-type theorem only from the condition of the generalized finite Dirichlet integral.

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arXiv2020

Asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second orderin unbounded domains

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a corollary we show also the Liouville-type theorem for both forward and ancient solutions.

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arXiv2020

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ and in the exterior domain outside of the large circle. The solution $v$ is handled in the class with $\nabla v \in L^q$ for $q \ge 2$. Since we deal with the case $q \ge 2$, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, i.e., for $q=2$ where a number of results such as asymptotic behavior of solutions have been observed. For the stationary problem we shall show that $ω(x)= o(|x|^{-(1/q + 1/q^2)})$ and $\nabla v(x) = o(|x|^{-(1/q+1/q^2)} \log |x|)$ as $|x| \to \infty$, where $ω\equiv {\rm rot\,} v$. As an application, we prove the Liouville type theorem under the assumption that $ω\in L^q(\mathbb{R}^2)$. For the non-stationary problem, a generalized $L^q$-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville type theorem for ancient solutions under the assumption that $ω\in L^q(\mathbb{R}^2 \times I)$.

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arXiv2017

Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains

Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

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