arXiv subjects

Mikihiro Fujii

Publications and source records attributed to Mikihiro Fujii.

26 records · Page 2Linked to original sources

arXiv2024

Large time behavior for solutions to the anisotropic Navier-Stokes equations in a 3D half-space

We consider the large time behavior of the solution to the anisotropic Navier--Stokes equations in a $3$D half-space. Investigating the precise anisotropic nature of linearized solutions, we obtain the optimal decay estimates for the nonlinear global solutions in anisotropic Lebesgue norms. In particular, we reveal the enhanced dissipation mechanism for the third component of velocity field. We notice that, in contrast to the whole space case, some difficulties arises on the $L^1(\mathbb{R}^3_+)$-estimates of the solution due to the nonlocal operators appearing in the linear solution formula. To overcome this, we introduce suitable Besov type spaces and employ the Littlewood--Paley analysis on the tangential space.

math.AP↗

arXiv2024

Well-posedness of the two-dimensional stationary Navier--Stokes equations around a uniform flow

In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is ill-posed for solutions around zero. In contrast, considering solutions around the non-zero constant flow, the perturbed system has a better regularity in the linear part, which enables us to prove the unique existence of solutions in the scaling critical spaces of the Besov type.

math.AP↗

arXiv2023

Ill-posedness of the two-dimensional stationary Navier--Stokes equations on the whole plane

We consider the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with $n \geqslant 3$, the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, despite the attention of many researchers, the corresponding problem in the two-dimensional whole plane case was a long-standing open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and prove the ill-posedness in the scaling critical Besov spaces based on $L^p(\mathbb{R}^2)$ for all $1 \leqslant p \leqslant2$ in the sense of the discontinuity of the solution map and the non-existence of small solutions. To overcome the difficulty, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier--Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed.

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arXiv2023

Counter examples for bilinear estimates related to the two-dimensional stationary Navier--Stokes equation

In this paper, we are concerned with bilinear estimates related to the two-dimensional stationary Navier--Stokes equation. By establishing concrete counter examples, we prove the bilinear estimates fail for almost all scaling critical Besov spaces. Our result may be closely related to an open problem whether the two-dimensional stationary Navier--Stokes equation on the whole plane $\mathbb{R}^2$ is well-posed or ill-posed in scaling critical Besov spaces.

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arXiv2023

Low Mach number limit of the global solution to the compressible Navier-Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier--Stokes system around the constant equilibrium states and prove the unique existence of a global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small and the incompressible part of the initial velocity generates the global solution of the incompressible Navier--Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier--Stokes equation in some space-time norms.

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arXiv2020

Isolated singularities in the heat equation behaving like fractional Brownian motions

We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2] $. It turns out that $H=1/N$ is critical.

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arXiv2020

Time periodic solutions to the $2$D quasi-geostrophic equation with the supercritical dissipation

We consider the $2$D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

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arXiv2019

Global solutions to the dissipative quasi-geostrophic equation with dispersive forcing

We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $κ(-Δ)^{α/2}θ (0<α\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a given initial data $θ_0$ which belongs to the scaling subcritical Sobolev space $H^s(\mathbb{R}^2)\ (s>2-α)$ if the size of dispersion parameter is sufficiently large. This phenomenon is so-called the global regularity. We also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. Moreover, we show the global regularity in the scaling critical Sobolev space $H^{2-α}(\mathbb{R}^2)$ and find that the size of dispersion parameter to ensure the global existence is determined by each subset $K\subset H^{2-α}(\mathbb{R}^2)$, which is precompact in some homogeneous Sobolev spaces.

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