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Mitsuo Higaki

Publications and source records attributed to Mitsuo Higaki.

18 recordsLinked to original sources
arXiv2026

A Runge-type theorem by remote forcing for the linearized resistive MHD system

In this paper, we study a quantitative Runge-type global approximation theorem for the linearized magnetohydrodynamic (MHD) system in bounded domains with arbitrary topology. In the context of magnetic relaxation, the interplay between the domain topology and magnetic field structure plays a crucial role. Recent studies illustrate a sharp contrast in the dynamics: while Enciso--Peralta-Salas (2025) highlights that the geometric complexity of magnetic fields acts as an obstruction to relaxation in non-resistive regimes, Kozono-Shimizu-Yanagisawa (2025) proves that in resistive regimes, the flow stably relaxes towards a harmonic equilibrium. Focusing on this resistive scenario, we adopt a control-theoretic viewpoint to quantitatively approximate the relaxation trajectory generated by the linearized initial-boundary value problem. Specifically, after decomposing the bounded-domain solution into the time-evolving part and the stationary part, we approximate it by a global solution on $\mathbb{R}^3$ under a remote forcing. An explicit dependence of the forcing cost on the approximation error is provided.

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arXiv2026

Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament

This paper investigates the dynamics of closed vortex filaments in $\R^3$ governed by the Localized Induction Equation. Recently, Aiki and Higaki (2026) established the nonlinear orbital stability of circular vortex filaments under asymmetric perturbations, while identifying Lyapunov instability due to the linear growth of translation modes. Motivated by this result, we prove the existence of a family of closed solutions, which we call axial screw motions, that bifurcate from a circular filament. These solutions remain uniformly close to the orbit of the circle, but drift secularly away from the reference motion because their translation speed along the symmetry axis differs from that of the circular filament. In particular, they provide explicit non-trivial perturbations that satisfy orbital-stability estimates while failing Lyapunov stability, thereby realizing the gap between orbital stability and Lyapunov stability near the circular filament.

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arXiv2026

Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations

We establish the nonlinear orbital stability of circular vortex filaments governed by the Localized Induction Equation (LIE) under non-symmetric perturbations, within the framework of [Tani-Nishiyama, 1997]. This result extends the first author's recent work [Aiki, 2025] by removing symmetry assumptions on perturbations. While the circular filaments are known to be Lyapunov unstable due to linear growth of the translational mode, we prove that their shape remains globally stable modulo spatial translations and rotations about the symmetry axis. The crucial ingredient is a geometric stability lemma derived from the conservation of vector fluid impulse, which constrains the low-frequency modulations that are not covered by the relative energy. Finally, we relate the fluid impulse to an isoperimetric inequality, yielding a geometric constraint for closed filaments.

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arXiv2025

An $L^2$-quantitative global approximation for the Stokes initial-boundary value problem

We establish the first quantitative Runge approximation theorem, with explicit $L^2$-estimates, for the 3d nonstationary Stokes system on a bounded spatial domain. This result addresses the two primary limitations of the qualitative result [H.-Sueur, 2025] obtained in collaboration with Franck Sueur: first, it bypasses the non-constructive Hahn-Banach theorem used in [H.-Sueur, 2025], precluding quantitative estimates; and second, it extends the scope of the theory from interior approximations to the physically important initial-boundary value problem. Our proof is founded on the modern quantitative framework of [R\"{u}land-Salo, 2019], which we adapt to the Stokes system by combining semigroup theory with a quantitative approximation for the associated resolvent problem.

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arXiv2025

Lagrangian controllability in perforated domains

The question at stake in Lagrangian controllability is whether one can move a patch of fluid particles to a target location by means of remote action in a given time interval. In the last two decades, positive results have been obtained both for the incompressible Euler and Navier-Stokes equations. However, for the latter, the case where the fluid is contained within domains bounded by solid boundaries with the no-slip condition has not been addressed, with respect to the difficulty caused by viscous boundary layers. In this paper, we investigate the Lagrangian controllability of viscous incompressible fluid in perforated domains for which the fraction of volume occupied by the holes is sufficiently small. Moreover, we quantitatively distinguish situations depending on the parameters for holes (diameter and distance) and for fluid (size of the initial data). Our approach relies on recent results on homogenization for evolutionary problems and on weak-strong stability estimates in measure of flows, alongside classical results on Runge-type approximations for elliptic equations and on Cauchy-Kowalevsky-type theorems for equations with analytic coefficients. Here, homogenization refers to the vanishing viscosity limit outside a porous medium, where (after scaling in time) the Navier-Stokes equations are homogenized to the Euler or Darcy equations. Indeed, in the proof, we act on the Navier-Stokes equations by strong and fast forcing to leverage inviscid approximations, which is a standard technique in the theory of controllability.

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arXiv2024

Wall laws for viscous flows in 3D randomly rough pipes: optimal convergence rates and stochastic integrability

This paper is concerned with effective approximations and wall laws of viscous laminar flows in 3D pipes with randomly rough boundaries. The random roughness is characterized by the boundary oscillation scale $\varepsilon \ll 1 $ and a probability space with ergodicity quantified by functional inequalities. The results in this paper generalize the previous work for 2D channel flows with random Lipschitz boundaries to 3D pipe flows with random boundaries of John type. Moreover, we establish the optimal convergence rates and substantially improve the stochastic integrability obtained in the previous studies. Additionally, we prove a refined version of the Poiseuille's law in 3D pipes with random boundaries, which seems unaddressed in the literature. Our systematic approach combines several classical and recent ideas (particularly from homogenization theory), including the Saint-Venant's principle for pipe flows, the large-scale regularity theory over rough boundaries, and applications of functional inequalities.

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arXiv2024

A Runge-type approximation theorem for the 3D unsteady Stokes system

We investigate Runge-type approximation theorems for solutions to the 3D unsteady Stokes system. More precisely, we establish that on any compact set with connected complement, local smooth solutions to the 3D unsteady Stokes system can be approximated with an arbitrary small positive error in $L^\infty$ norm by a global solution of the 3D unsteady Stokes system, where the velocity grows at most exponentially at spatial infinity and the pressure grows polynomially. Additionally, by considering a parasitic solution to the Stokes system, we establish that some growths at infinity are indeed necessary.

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arXiv2024

Axisymmetric steady Navier-Stokes flows under suction

We prove the existence of solutions for the axisymmetric steady Navier-Stokes system around an infinite cylinder under external forces. The solutions are constructed to be decaying at the horizontal infinity, despite an analogue of the Stokes paradox for the linearized system, and having neither periodicity nor decay in the vertical direction. The proof is based on perturbation of the nonlinear system around a suction flow. The class of functions in this paper, which is a subspace of the space of Fourier transformed vector finite Radon measures, is inspired by Giga-Saal (2013) treating rotating boundary layers.

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arXiv2023

Existence of steady Navier-Stokes flows exterior to an infinite cylinder

We consider the three-dimensional steady Navier-Stokes system in the exterior of an infinite cylinder under the action of an external force. We construct solutions in the class of vertically uniform flows which vanish at horizontal infinity. More precisely, for a boundary datum determined by a rotating flow and a suction flow, and for a small force of the form $f=g+\operatorname{div} F$ with suitable decay, we prove the existence of a weak solution asymptotic to the corresponding Hamel-type flow. Although all data are independent of the vertical variable, the problem is not reduced to the planar exterior Navier-Stokes system: the vertical component satisfies a separate transport-diffusion equation involving the two-dimensional Laplacian, whose fundamental solution has logarithmic growth. The proof is based on a mode-by-mode analysis of the linearized three-dimensional problem around the Hamel-type flow and a contraction argument.

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arXiv2023

Stability of planar exterior stationary flows with suction

We consider the two-dimensional Navier-Stokes system in a domain exterior to a disk. The system admits a stationary solution with critical decay $O(|x|^{-1})$ written as a linear combination of the pure rotating flow and the flux carrier. We prove its nonlinear stability in large time for initial disturbances in $L^2$ under smallness conditions, assuming that there is suction across the boundary, namely that the sign of coefficients of the flux carrier is negative. This result partially solves an open problem in the literature.

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arXiv2022

Existence of planar non-symmetric stationary flows with large flux in an exterior disk

This paper is concerned with the two-dimensional stationary Navier-Stokes system in the domain exterior to the unit disk. The existence of solutions with critical decay $O(|x|^{-1})$ is established around some explicit flows with large flux. The solutions are obtained for non-symmetric external forces, and moreover, are unique in a certain class.

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arXiv2022

Higher-order boundary layers and regularity for Stokes systems over rough boundaries

In this paper, we study the large-scale boundary regularity for the Stokes system in periodically oscillating John domains. Our main contribution is the construction of boundary layer correctors of arbitrary order. This is a significant generalization of the known results restricted to the first and second orders. As an application, we prove the large-scale regularity estimate, as well as a Liouville theorem, of arbitrary order for the Stokes system. Our results are also related to higher-order boundary layer tails and wall laws in viscous fluids over rough boundaries.

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arXiv2021

Large-scale regularity for the stationary Navier-Stokes equations over non-Lipschitz boundaries

In this paper we address the large-scale regularity theory for the stationary Navier-Stokes equations in highly oscillating bumpy John domains. These domains are very rough, possibly with fractals or cusps, at the microscopic scale, but are amenable to the mathematical analysis of the Navier-Stokes equations. We prove: (i) a large-scale Calder\'on-Zygmund estimate, (ii) a large-scale Lipschitz estimate, (iii) large-scale higher-order regularity estimates, namely, $C^{1,\gamma}$ and $C^{2,\gamma}$ estimates. These nice regularity results are inherited only at mesoscopic scales, and clearly fail in general at the microscopic scales. We emphasize that the large-scale $C^{1,\gamma}$ regularity is obtained by using first-order boundary layers constructed via a new argument. The large-scale $C^{2,\gamma}$ regularity relies on the construction of second-order boundary layers, which allows for certain boundary data with linear growth at spatial infinity. To the best of our knowledge, our work is the first to carry out such an analysis. In the wake of many works in quantitative homogenization, our results strongly advocate in favor of considering the boundary regularity of the solutions to fluid equations as a multiscale problem, with improved regularity at or above a certain scale.

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arXiv2019

Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law

We investigate regularity estimates for the stationary Navier-Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved $C^{1,μ}$ estimate and identify the building blocks of the regularity theory, dubbed `Navier polynomials'. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier-Stokes equations, our results do not require any smallness assumption on the solutions.

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arXiv2019

Note on the stability of planar stationary flows in an exterior domain without symmetry

The asymptotic stability of two-dimensional stationary flows in a non-symmetric exterior domain is considered. Under the smallness condition on initial perturbations, we show the stability of the small stationary flow whose leading profile at spatial infinity is given by the rotating flow decaying in the scale-critical order $O(|x|^{-1})$. Especially, we prove the $L^p$-$L^q$ estimates to the semigroup associated with the linearized equations.

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arXiv2017

On stationary two-dimensional flows around a fast rotating disk

We study the two-dimensional stationary Navier-Stokes equations describing flows around a rotating disk. The existence of unique solutions is established for any rotating speed, and qualitative effects of a large rotation are described precisely by exhibiting a boundary layer structure and an axisymmetrization of the flow.

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