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Yutaka Terasawa

Publications and source records attributed to Yutaka Terasawa.

14 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↗
arXiv2025

Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels.

math.AP↗
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.

math.AP↗
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.

math.AP↗
arXiv2022

Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities

We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.

math.AP↗
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.

math.AP↗
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 $ω\in L^q(\mathbb{R}^2 \times I)$.

math.AP↗
arXiv2019

Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities and Nonlocal Free Energies

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the $H^{α/2}$-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.

math.AP↗
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.

math.AP↗
arXiv2013

Existence of Weak Solutions for a Diffuse Interface Model of Non-Newtonian Two-Phase Flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of Non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.

math.AP↗
arXiv2012

Positive operators and maximal operators in a filtered measure space

In a filtered measure space, a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete Wolff's potential. A refinement of the Carleson embedding theorem is also introduced. Sawyer type characterization of weights for which a two-weight norm inequality for a generalized Doob's maximal operator holds is established by an application of our Carleson embedding theorem. Moreover, Hytönen-Pérez type one-weight norm estimate for Doob's maximal operator is obtained by the use of our two-weight characterization.

math.CA↗
arXiv2012

Stochastic power law fluids: Existence and uniqueness of weak solutions

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force. We investigate the existence and the uniqueness of weak solutions to this SPDE.

math.PR↗