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Mikihiro Fujii

Publications and source records attributed to Mikihiro Fujii.

At least 19 recordsLinked to original sources
arXiv2026

Refined decay estimates for global solutions to the rotating MHD equations

In this paper, we consider the Cauchy problem for the incompressible magnetohydrodynamic equations with the Coriolis force in the three-dimensional whole space. For large initial data, it is known that the Cauchy problem admits a unique global solution in critical Sobolev space $\dot{H}^{1/2}(\mathbb{R}^3)$ provided that the speed of rotation is fast enough. By using delicate dispersive estimates, we show refined decay estimates of the velocity field and magnetic field. These decay rates significantly improve the ones obtained by Kim [\emph{J. Differential Equations.}, 2022] in the subcritical Sobolev framework $H^s(\mathbb{R}^3)$ with $1/2<s<3/2$.

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arXiv2026

Global solutions to the Navier--Stokes equations with large vertical velocities in $\dot{B}_{\infty,σ}^{-1}(\mathbb{R}^3)$

In this paper, we consider the Cauchy problem for the $3$D incompressible Navier--Stokes equations and prove the existence of unique global solutions in the framework that the horizontal component of the velocity field is small in some critical Besov spaces including the classical Fujita--Kato class, while the vertical component is large in the wide class $\dot{B}_{\infty,σ}^{-1}(\mathbb{R}^3)$ ($1 \leq σ< \infty$) where the Navier--Stokes equations are known to be ill-posed in.

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arXiv2026

Sharp decay estimates for global solutions to the incompressible rotating Navier--Stokes equations

In this paper, we consider the three-dimensional incompressible rotating Navier--Stokes equations and establish the sharp $L^p$ decay estimates of global solutions. We reveal that the optimal $L^p$ decay rates for $2<p<\infty$ are strictly faster than those obtained in existing results by interpolation between the $L^2$ unitary identity and $L^\infty$ dispersive estimates, although the endpoint cases were known to be sharp. Moreover, the optimality of decay rates is also proved by the lower bound estimate for a specific initial datum. The underlying mechanism lies in the anisotropic degeneracy of the oscillatory integrals arising from the Coriolis force.

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arXiv2026

Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces

It is known that uniqueness of mild solutions to the incompressible Navier-Stokes equations holds in the critical class $C([0,T);L^n(\mathbb{R}^n))$ for $n \geqslant 3$. In this paper, we prove that this result is sharp in the sense that uniqueness fails if $L^n(\mathbb{R}^n)$ is replaced by some scaling critical spaces that are even slightly larger. We achieve this through a complete classification for every pair $(p,q)$ of whether uniqueness of mild solutions in the critical Besov class $C([0,T);\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n))$ holds or not. Our non-uniqueness mechanism produces infinitely many global solutions emanating even from zero initial state, whose large-time asymptotics are governed by non-trivial stationary flow. To the best of our knowledge, such non-unique solutions provide the first examples of non-dissipative unforced Navier-Stokes flow with critical regularity.

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arXiv2026

Global strong solutions to the compressible Navier--Stokes--Coriolis system for large data

We consider the compressible Navier--Stokes system with the Coriolis force on the $3$D whole space. In this model, the Coriolis force causes the linearized solution to behave like a $4$th order dissipative semigroup $\{ e^{-tΔ^2} \}_{t>0}$ with slower time decay rates than the heat kernel, which creates difficulties in nonlinear estimates in the low-frequency part and prevents us from constructing the global strong solutions by following the classical method. On account of this circumstance, the existence of unique global strong solutions has been open even in the classical Matsumura--Nishida framework. In this paper, we overcome the aforementioned difficulties and succeed in constructing a unique global strong solution in the framework of scaling critical Besov spaces. Furthermore, our result also shows that the global solution is constructed for arbitrarily large initial data provided that the speed of the rotation is high and the Mach number is low enough by focusing on the dispersive effect due to the mixture of the Coriolis force and acoustic wave.

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arXiv2025

Analyticity in space and time for global solutions to the anisotropic Navier--Stokes equations in the critical $L^p(\mathbb{R}^3)$ framework

In the present paper, we consider the real analyticity of the global solutions to the $3$D incompressible anisotropic Navier--Stokes equations. We show that if only the horizontal component of initial velocity is small and analytic in $x_3$, then there exists a unique global solution which is analytic in $t>0$ and $x\in \mathbb{R}^3$. Our functional framework lies in some anisotropic Besov spaces based on $L^p(\mathbb{R}^3)$. To our best knowledge, this paper is the first contribution to the well-posedness of the anisotropic Navier--Stokes equations in function spaces of the Besov type based on the full $L^p(\mathbb{R}^3)$ setting.

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arXiv2025

Asymptotic instability for the forced Navier--Stokes equations in critical Besov spaces

The asymptotic stability is one of the classical problems in the field of mathematical analysis of fluid mechanics. In $\mathbb{R}^n$ with $n \geq 3$, it is easily proved by the standard argument that if the given small external force decays at temporal infinity, then the small forced Navier--Stokes flow also strongly converges to zero as time tends to infinity in the framework of the critical Besov spaces $\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$ with $1 \leq p < n$ and $1 \leq q < \infty$. In the present paper, we show that this asymptotic stability fails for $p \geq n$ with $n \geq 3$ in the sense that there exist arbitrary small external forces whose critical Besov norm decays in large time, whereas the corresponding Navier--Stokes flows oscillate and do not strongly converge as $t \to \infty$ in the framework of the critical Besov spaces $\dot{B}_{p,q}^{n/p-1}(\mathbb{R}^n)$. Moreover, we find that the situation is different in the two-dimensional case $n=2$ and show the forced Navier--Stokes flow is asymptotically unstable in $\dot{B}_{p,1}^{2/p-1}(\mathbb{R}^2)$ for all $1 \leq p \leq \infty$. Our instability does not appear in the linear level but is caused by the nonlinear interaction from external forces.

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arXiv2025

Time-periodic solutions to the Navier--Stokes equations on the whole space including the two-dimensional case

Let us consider the incompressible Navier--Stokes equations with the time-periodic external forces in the whole space $\mathbb{R}^n$ with $n\geq 2$ and investigate the existence and non-existence of time-periodic solutions. In the higher dimensional case $n \geq 3$, we construct a unique small solution for given small time-periodic force in the scaling critical spaces of Besov type and prove its stability under small perturbations. In contrast, for the two-dimensional case $n=2$, the time-periodic solvability of the Navier--Stokes equations has been long standing open. It is the central work of this paper that we have now succeeded in solving this issue negatively by providing examples of small external forces such that each of them does not generate time-periodic solutions.

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arXiv2025

Global strong solutions to the $3$D rotating compressible Navier--Stokes--Korteweg system for large data in the critical $\widehat{L^p}$ framework

Let us consider the $3$D compressible Navier--Stokes--Korteweg system in the rotational framework. Although there is a wealth of literature on the weak solutions to this system, there seem to be no results on the strong solutions. In this paper, we show the unique existence of global solutions for {\it large} initial data in the critical Besov-type spaces based on the Fourier--Lebesgue spaces $\widehat{L^p}(\mathbb{R}^3)$ with $2 \leq p < 3$, provided that the rotation speed and the Mach number are sufficiently large and small, respectively. The key ingredient of the proof is to establish the Strichartz-type estimates due to the dispersion caused by the mixture of the rotation and acoustic waves in the Fourier--Lebesgue spaces, and focus on the better structure of dissipation from the Korteweg term and the nonlinear terms of the divergence form in the momentum formulation.

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arXiv2025

Decay rates of three dimensional stationary Navier--Stokes flows at the spatial infinity

In this paper, we establish the well-posedness results of the three dimensional stationary Navier--Stokes equations (SNS) in some critical hybrid type Besov spaces with respect to the scaling invariant structure of (SNS). Although such critical functional spaces contain the functions with singularities, we give some sufficient conditions such that the $L^{\infty}$-norm of the solutions of (SNS) decay at the infinity within some polynomial type rate.

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arXiv2025

Compressible Navier--Stokes--Coriolis system in critical Besov spaces

We consider the three-dimensional compressible Navier--Stokes system with the Coriolis force and prove the long-time existence of a unique strong solution. More precisely, we show that for any $0<T<\infty$ and arbitrary large initial data in the scaling critical Besov spaces, the solution uniquely exists on $[0,T]$ provided that the speed of rotation is high and the Mach numbers are low enough. To the best of our knowledge, this paper is the first contribution to the well-posedness of the \textit{compressible} Navier--Stokes system with the Coriolis force in the whole space $\mathbb R^3$. The key ingredient of our analysis is to establish the dispersive linear estimates despite a quite complicated structure of the linearized equation due to the anisotropy of the Coriolis force.

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arXiv2025

Incompressible and fast rotation limits for 3D compressible rotating Euler system with general initial data

This paper is concerned with the low Mach and Rossby number limits of $3$D compressible rotating Euler equations with ill-prepared initial data in the whole space. More precisely, the initial data is the sum of a $3$D part and a $2$D part. With the help of a suitable intermediate system, we perform this singular limit rigorously with the target system being a $2$D QG-type. This particularly gives an affirmative answer to the question raised by Ngo and Scrobogna [\emph{Discrete Contin. Dyn. Syst.}, 38 (2018), pp. 749-789]. As a by-product, our proof gives a rigorous justification from the $2$D inviscid rotating shallow water equations to the $2$D QG equations in whole space.

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arXiv2025

Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation

We consider the stationary problem for the quasi-geostrophic equation on the whole plane and investigate its well-posedness and ill-posedness. In[Fujii, Ann. PDE 10, 10 (2024)], it was shown that the two-dimensional stationary Navier--Stokes equations are ill-posed in the critical Besov spaces $\dot B_{p,1}^{\frac{2}{p}-1}(\mathbb{R}^2)$ with $1 \leq p \leq 2$. Although the quasi-geostrophic equation has the same invariant scale structure as the Navier--Stokes equations, we reveal that the quasi-geostrophic equation is well-posed in the scaling critical Besov spaces $\dot B_{p,q}^{\frac{2}{p}-1}(\mathbb{R}^2)$ with $(p,q) \in [1,4) \times [1,\infty]$ or $(p,q)=(4,2)$ due to the better properties of the nonlinear structure of the quasi-geostrophic equation compared to that of the Navier--Stokes equations. Moreover, we also prove the optimality for the above range of $(p,q)$ ensuring the well-posedness in the sense that the stationary quasi-geostrophic equation is ill-posed for all the other cases.

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arXiv2025

Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space

We consider the stationary problem for the quasi-geostrophic equation with the critical and super-critical dissipation and prove the unique existence of small solutions for given small external force in the scaling critical Sobolev spaces framework. Moreover, we also show that the data-to-solution map is continuous. Since the critical and super-critical case involves the derivative loss, which affects the class of the continuity of the data-to-solution map, we reveal that the map is no longer uniform continuous, in contrast to the sub-critical case, where the Lipschitz continuity holds.

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arXiv2024

Stationary Navier--Stokes equations on the half spaces in the scaling critical framework

In this paper, we consider the inhomogeneous Dirichlet boundary value problem for the stationary Navier--Stokes equations in $n$-dimensional half spaces $\mathbb{R}^n_+= \{ x=(x',x_n)\ ;\ x' \in \mathbb{R}^{n-1}, x_n > 0 \}$ with $n \geq 3$ and prove the well-posedness in the scaling critical Besov spaces. Our approach is to regard the system as an evolution equation for the normal variable $x_n$ and reformulate it as an integral equation. Then, we achieve the goal by making use of the maximal regularity method that has developed in the context of nonstationary analysis in critical Besov spaces. Furthermore, for the case of $n \geq 4$, we find that the asymptotic profile of the solution as $x_n \to \infty$ is given by the $(n-1)$-dimensional stationary Navier--Stokes flow.

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arXiv2024

Low Mach number limit for the global large solutions to the $2$D Navier--Stokes--Korteweg system in the critical $\widehat{L^p}$ framework

In the present paper, we consider the compressible Navier--Stokes--Korteweg system on the $2$D whole plane and show that a unique global solution exists in the scaling critical Fourier--Besov spaces for arbitrary large initial data provided that the Mach number is sufficiently small. Moreover, we also show that the global solution converges to the $2$D incompressible Navier--Stokes flow in the singular limit of zero Mach number. The key ingredient of the proof lies in the nonlinear stability estimates around the large incompressible flow via the Strichartz estimate for the linearized equations in Fourier--Besov spaces.

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arXiv2024

Linear and nonlinear stability for the $3$D stratified Boussinesq equations with the horizontal viscosity and diffusivity

In this manuscript, we consider the $3$D Boussinesq equations for stably stratified fluids with the horizontal viscosity and thermal diffusivity and investigate the large time behavior of the solutions. Making use of the anisotropic Littlewood--Paley theory, we obtain their precise $L^1$-$L^p$ decay estimates, which provide us information on both the anisotropic and dispersive structure of the system. More precisely, we reveal that the dispersion from the skew symmetric terms of stratification makes the decay rates of some portions of the solutions faster and furthermore the third component of the velocity field exhibit the enhanced dissipative effect, which provides the additional fast decay rate.

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