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Yasunori Maekawa

Publications and source records attributed to Yasunori Maekawa.

At least 19 recordsLinked to original sources
arXiv2026

Local separation profiles for the stationary Prandtl system

We construct solutions to the two-dimensional stationary Prandtl system under a constant adverse pressure gradient exhibiting a family of distinct separation laws. More precisely, for every integer $\ell\ge1$, we construct smooth initial data for which separation occurs at a finite location $x=x^\ast$, with \[ \partial_y u(x,0) =\mathbf C_\ell(x^\ast-x)^{\ell/2}\bigl(1+o(1)\bigr), \qquad x\uparrow x^\ast, \] where $\mathbf C_\ell>0$ depends on the initial data. We determine the local asymptotic structure of the solution near separation through matched inner and outer expansions based on linearization around the ground state in von Mises variables. This description addresses Open Problem~5 posed by Oleinik and Samokhin~\cite{OleinikSamok-book-99}. We also prove that the square-root separation regime, corresponding to $\ell=1$, is dynamically stable under small admissible perturbations of the initial data in a suitable topology.

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arXiv2025

Self-similar vorticity around the boundary and non-uniqueness of solutions to the two-dimensional Navier-Stokes equations in the half space

In this paper we show the non-uniqueness of mild solutions to the two-dimensional forced Navier-Stokes equations in the half space under the noslip boundary condition, following the program established by Albritton, Bru{é}, and Colombo in 2022. Our construction of non-unique solutions is based on the instability of self-similar vorticity at high Reynolds numbers which concentrates around the boundary at the initial time. In our construction, therefore, a kind of boundary layer has to be taken into account in the analysis, contrasting to the known results where the unstable self-similar vorticity is located away from the boundary with $O(1)$ distance around the initial time.

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arXiv2024

Local Rigidity of the Couette Flow for the Stationary Triple-Deck Equations

The Triple-Deck equations are a classical boundary layer model which describes the asymptotics of a viscous flow near the separation point, and the Couette flow is an exact stationary solution to the Triple-Deck equations. In this paper we prove the local rigidity of the Couette flow in the sense that there are no other stationary solutions near the Couette flow in a scale invariant space. This provides a stark contrast to the well-studied stationary Prandtl counterpart, and in particular offers a first result towards the rigidity question raised by R. E. Meyer in 1983.

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arXiv2023

Optimal rate of convergence to nondegenerate asymptotic profiles for fast diffusion in domains

This paper is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded domains, where every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile. Bonforte and Figalli (CPAM, 2021) first proved an exponential convergence to nondegenerate positive asymptotic profiles for nonnegative rescaled solutions in a weighted $L^2$ norm for smooth bounded domains by developing a nonlinear entropy method. However, the optimality of the rate remains open to question. In the present paper, their result is fully extended to possibly sign-changing asymptotic profiles as well as general bounded domains by improving an energy method along with a quantitative gradient inequality developed by the first author (ARMA, 2023). Moreover, a (quantitative) exponential stability result for least-energy asymptotic profiles follows as a corollary, and it is further employed to prove the optimality of the exponential rate.

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arXiv2022

Improved well-posedness for the Triple-Deck and related models via concavity

We establish linearized well-posedness of the Triple-Deck system in Gevrey-$\frac32$ regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result \cite{DietertGV}, one cannot expect a generic improvement of the result of \cite{IyerVicol} to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier-Laplace side: i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from \cite{GVMM} ii) smoothing properties of the Benjamin-Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier-Stokes equations: we show for this system a similar Gevrey-$\frac32$ linear well-posedness result for concave data, improving at the linear level the recent work \cite{MR4149066}.

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arXiv2021

Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Prüss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate $O(e^{-\sqrtν\,t})$ when the viscosity coefficient $ν$ is sufficiently small as in the case of the plane Kolmogorov flow.

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arXiv2020

Optimal Prandtl expansion around concave boundary layer

We provide an optimal Gevrey stability result for general boundary layer expansions, under a mild concavity condition on the boundary layer profile. Our result generalizes (and even improves in the non strictly concave case) the one obtained in [Gerard-Varet et al, Duke Math. J. 167 (2018)], restricted to expansions of shear flow type.

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arXiv2019

On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $ν$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(ν^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(ν^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.

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arXiv2019

On stability of blow-up solutions of the Burgers vortex type for the Navier-Stokes equations with a linear strain

We study the three-dimensional Navier-Stokes equations in the presence of the axisymmetric linear strain, where the strain rate depends on time in a specific manner. It is known that the system admits solutions which blow up in finite time and whose profiles are in a backward self-similar form of the familiar Burgers vortices. In this paper it is shown that the existing stability theory of the Burgers vortex leads to the stability of these blow-up solutions as well. The secondary blow-up is also observed when the strain rate is relatively weak.

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arXiv2019

Estimates for the Navier-Stokes equations in the half-space for non localized data

This paper is devoted to the study of the Stokes and Navier-Stokes equations, in a half-space, for initial data in a class of locally uniform Lebesgue integrable functions, namely $L^q_{uloc,σ}(\R^d_+)$. We prove the analyticity of the Stokes semigroup $e^{-t{\bf A}}$ in $L^q_{uloc,σ}(\R^d_+)$ for $1<q\leq\infty$. This follows from the analysis of the Stokes resolvent problem for data in $L^q_{uloc,σ}(\R^d_+)$, $1<q\leq\infty$. We then prove bilinear estimates for the Oseen kernel, which enables to prove the existence of mild solutions. The three main original aspects of our contribution are: (i) the proof of Liouville theorems for the resolvent problem and the time dependent Stokes system under weak integrability conditions, (ii) the proof of pressure estimates in the half-space and (iii) the proof of a concentration result for blow-up solutions of the Navier-Stokes equations. This concentration result improves a recent result by Li, Ozawa and Wang and provides a new proof.

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arXiv2019

Local energy weak solutions for the Navier-Stokes equations in the half-space

The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemarié-Rieusset solutions in the whole space $\mathbb R^3$. The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical $L^3(\mathbb R^3_+)$ norm obtained by Barker and Seregin for solutions developing a singularity in finite time.

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arXiv2018

Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations

We show the $H^1$ stability of shear flows of Prandtl type: $U^ν= (U_s(y/\sqrtν),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.

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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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arXiv2017

On convergence criteria for incompressible Navier-Stokes equations with Navier boundary conditions and physical slip rates

We prove some criteria for the convergence of weak solutions of the 2D incompressible Navier-Stokes equations with Navier slip boundary conditions to a strong solution of incompressible Euler. The slip rate depends on a power of the Reynolds number, and it is increasingly apparent that the power 1 may be critical for L^2 convergence, as hinted at in [hal-01093331].

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arXiv2016

Existence and stability of viscous vortices

Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real flows vortex lines are usually not straight, and can even form closed curves, as in the case of axisymmetric vortex rings which are very common in nature and in laboratory experiments. The aim of this chapter is to review a few rigorous results concerning existence and stability of viscous vortices in simple geometries.

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