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We consider ($-\alpha$)-homogeneous solutions (stationary self-similar solutions of degree $-\alpha$) to the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions., Comment: 52 pages

We consider ($-\alpha$)-homogeneous solutions to the stationary incompressible Euler equations in $\mathbb{R}^{3}\backslash\{0\}$ for $\alpha\geq 0$ and in $\mathbb{R}^{3}$ for $\alpha<0$. Shvydkoy (2018) demonstrated the nonexistence of ($-1$)-homogeneous solutions and ($-\alpha$)-homogeneous solutions in the range $0\leq \alpha\leq 2$ for the Beltrami and axisymmetric flows. The nonexistence result of the Beltrami ($-\alpha$)-homogeneous solutions holds for all $\alpha<1$. We show the nonexistence of axisymmetric ($-\alpha$)-homogeneous solutions without swirls for $-2\leq \alpha<0$. The main result of this study is the existence of axisymmetric ($-\alpha$)-homogeneous solutions in the complementary range $\alpha\in \mathbb{R}\backslash [0,2]$. More specifically, we show the existence of axisymmetric Beltrami ($-\alpha$)-homogeneous solutions for $\alpha\in \mathbb{R}\backslash [0,2]$ and axisymmetric ($-\alpha$)-homogeneous solutions with a nonconstant Bernoulli function for $\alpha\in \mathbb{R}\backslash [-2,2]$. This is the first existence result on ($-\alpha$)-homogeneous solutions with no explicit forms. For $2<\alpha<3$, constructed ($-\alpha$)-homogeneous solutions provide new examples of the Beltrami/Euler flows in $\mathbb{R}^{3}\backslash\{0\}$ whose level sets of the proportionality factor/Bernoulli surfaces are nested surfaces created by the rotation of the sign $``\infty"$., Comment: 69 pages

Chandrasekhar's nonlinear force-free field, discovered in 1956, is a rare example of a nonlinear force-free field describing a large coherent structure of axisymmetric magnetic field lines with swirls in the presence of Alfv\'en waves. We demonstrate that his traveling wave solution is orbitally stable with the energy norm in ideal MHD regarding weak ideal limits of axisymmetric Leray-Hopf solutions., Comment: 85 pages, new remark 1.3 and Appendix B

We prove that bounded Beltrami fields must be symmetric if a proportionality factor depends on 2 variables in the cylindrical coordinate and admits a regular level set diffeomorphic to a cylinder or a torus., Comment: 12 pages

We construct traveling wave solutions to the 3d Euler equations by axisymmetric Beltrami fields with a non-constant proportionality factor. They form a vortex ring with nested invariant tori consisting of level sets of the proportionality factor., Comment: 30 pages

We prove that the Stokes semigroup is a bounded analytic semigroup on $L^{\infty}_{\sigma}$ of angle $\pi/2$ for two-dimensional exterior domains. This result is an end point case of the $L^{p}$-boundedness of the semigroup for $p\in (1,\infty)$, established by Borchers and Varnhorn (1993) and an extension of finite time $L^{\infty}$-estimates studied by the author and Giga (2014). The proof is based on the non-existence result of bounded steady flows (the Stokes paradox) and some asymptotic formula for the net force of the Stokes resolvent., Comment: 20 pages

The Lamb dipole is a traveling wave solution to the two-dimensional Euler equations introduced by S. A. Chaplygin (1903) and H. Lamb (1906) at the early 20th century. We prove orbital stability of this solution based on a vorticity method initiated by V. I. Arnold. Our method is a minimization of a penalized energy with multiple constraints that deduces existence and orbital stability for a family of traveling waves. As a typical case, orbital stability of the Lamb dipole is deduced by characterizing a set of minimizers as an orbit of the dipole by a uniqueness theorem in the variational setting., Comment: 41 pages

We consider the two-dimensional Navier-Stokes equations subject to the Dirichlet boundary condition in a half plane for initial vorticity with finite measures. We study local well-posedness of the associated vorticity equations for measures with a small pure point part and global well-posedness for measures with a small total variation. Our construction is based on an $L^{1}$-estimate of a solution operator for the vorticity equations associated with the Stokes equations., Comment: 48 pages

We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup on $L^{ \infty}_{ \sigma}$ of angle $ \pi/2$ for $n \geq 3$. We also prove large time estimates for $n=2$ with zero net force., Comment: 26 pages

We prove unique existence of local-in-time smooth solutions of the Navier-Stokes equations for initial data in $L^{p}$ and $p \in [3, \infty)$ in an infinite cylinder, subject to the Neumann boundary condition., Comment: 28 pages, final version

We construct global weak solutions of the Euler equations in an infinite cylinder $\Pi=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for axisymmetric initial data without swirl when initial vorticity $\omega_{0}=\omega^{\theta}_{0}e_{\theta}$ satisfies $\omega^{\theta}_{0}/r\in L^{q}$ for $q\in [3/2,3)$. The solutions constructed are H\"older continuous for spatial variables in $\overline{\Pi}$ if in addition that $\omega^{\theta}_{0}/r\in L^{s}$ for $s\in (3,\infty)$ and unique if $s=\infty$. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-posed for axisymmetric data without swirl in $L^{p}$ for all $p\in [3,\infty)$. It is also shown that the energy dissipation tends to zero if $\omega^{\theta}_{0}/r\in L^{q}$ for $q\in [3/2,2]$, and Navier-Stokes flows converge to Euler flow in $L^{2}$ locally uniformly for $t\in [0,\infty)$ if additionally $\omega^{\theta}_{0}/r\in L^{\infty}$. The $L^{2}$-convergence in particular implies the energy equality for weak solutions., Comment: 37 pages. The title is changed from the previous version. Remarks 6.4 and new references are added

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi)$ satisfying the decay condition of the swirl component $ru^{\theta}_{0}\in L^{\infty}(\Pi)$., Comment: Final version. The title is changed from "Axisymmetric flows in an exterior domain"

We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers., Comment: 40 pages, final version

We prove unique existence of mild solutions on $L^{\infty}_{\sigma}$ for the Navier-Stokes equations in an exterior domain in $ \mathbb{R}^{n}$, $n\geq 2$, subject to the non-slip boundary condition., Comment: 17 pages, final version

In this paper, we study regularizing effects of the composition operator $S(t)\mathbb{P}\partial$ for the Stokes semigroup $S(t)$ and the Helmholtz projection $\mathbb{P}$ in a space of bounded functions. We establish new a priori $L^{\infty}$-estimates of the operator $S(t)\mathbb{P}\partial$ for a certain class of domains including bounded and exterior domains. They imply unique existence of mild solutions of the Navier-Stokes equations in a space of bounded functions., Comment: 35 pages, final version

The Stokes equation on a domain $\Omega \subset R^n$ is well understood in the $L^p$-setting for a large class of domains including bounded and exterior domains with smooth boundaries provided $1