arXiv subjects

Xukai Yan

Publications and source records attributed to Xukai Yan.

13 recordsLinked to original sources

arXiv2026

Classification of local $(-1)$-homogeneous axisymmetric solutions of the $3$D stationary Navier-Stokes equations with logarithmic singular-ray behavior

We study the existence and classification of local $(-1)$-homogeneous axisymmetric solutions $(u, p)$ of the three-dimensional incompressible stationary Navier-Stokes equations near a singular ray. We focus on such solutions with Type II behavior, which satisfy $0<\limsup_{x\in\bS^2, x\to P}|u|/|\ln \text{dist}(x, P)|<\infty$, where $P$ is the south or north pole. Apart from the Landau solutions, these are the least singular nontrivial $(-1)$-homogeneous axisymmetric solutions. We construct a four-parameter family of local $(-1)$-homogeneous axisymmetric solutions with at most logarithmic growth near the singular ray and represent them as convergent series whose coefficients are determined recursively. Conversely, we prove that every local $(-1)$-homogeneous axisymmetric solution satisfying $|u|=o(\text{dist} (x, P)^{-1})$ as $x\to P$ on $\bS^2$ belongs to this family. In particular, this family contains Type II solutions with nonzero swirl, in contrast to the global $(-1)$-homogeneous axisymmetric Type II solutions in $\R^3\setminus\{x'=0\}$, which have no swirl. We also establish derivative estimates for the constructed solutions and identify the singular force generated by these solutions across the singular ray in the sense of distributions.

math.AP↗

arXiv2025

Rigidity for homogeneous solutions to the two-dimensional Euler equations in sector-type domains

We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a 0$, and $b = +\infty$ or $b < +\infty$, we show that if a solution satisfies some homogeneity assumptions on the boundary of $\Omega_{a, b, \theta_0}$ and if the radial or angular component of the velocity does not vanish in $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$, then it must be homogeneous throughout $\overline{\Omega_{a, b, \theta_0}}\setminus\{\bm{0}\}$.

math.AP↗

arXiv2025

Recent research on $(-1)$-homogeneous solutions of stationary Navier-Stokes equations

We make an exposition of recent research on $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We also discuss properties of such solutions that are axisymmetric with no swirl, and present graphs illustrating examples that exhibit various typical types of singular behavior.

math.AP↗

arXiv2024

Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations

We study the removable singularity problem for $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local $(-1)$-homogeneous solution $u$ near a potential singular ray from the origin, which passes through a point $P$ on the unit sphere $\mathbb{S}^2$, can be smoothly extended across $P$ on $\mathbb{S}^2$, provided that $u=o(\ln \text{dist} (x, P))$ on $\mathbb{S}^2$. The result is optimal in the sense that for any $\alpha>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-\alpha$. Furthermore, we discuss the behavior of isolated singularities of $(-1)$-homogeneous solutions and provide examples from the literature that exhibit varying behaviors. We also present an existence result of solutions with any finite number of singular points located anywhere on $\mathbb{S}^2$.

math.AP↗

arXiv2021

Anisotropic Caffarelli-Kohn-Nirenberg type inequalities

Caffarelli, Kohn and Nirenberg considered in 1984 the interpolation inequalities \[\||x|^{γ_1}u\|_{L^s(\mathbb{R}^n)}\le C\||x|^{γ_2}\nabla u\|_{L^p(\mathbb{R}^n)}^a\||x|^{γ_3}u\|_{L^q(\mathbb{R}^n)}^{1-a} \] in dimension $n\ge 1$, and established necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Motivated by our study of the asymptotic stability of solutions to the Navier-Stokes equations, we consider a more general and improved anisotropic version of the interpolation inequalities \[ \||x|^{γ_1}|x'|^αu\|_{L^s(\mathbb{R}^n)}\le C\||x|^{γ_2}|x'|^μ\nabla u\|_{L^p(\mathbb{R}^n)}^{a}\||x|^{γ_3}|x'|^βu\|_{L^q(\mathbb{R}^n)}^{1-a} \] in dimensions $n\ge 2$, where $x=(x', x_n)$ and $x'=(x_1, ..., x_{n-1})$, and give necessary and sufficient conditions for which to hold under natural assumptions on the parameters. Moreover we extend the Caffarelli-Kohn-Nirenberg inequalities from $q\ge 1$ to $q>0$. This extension, together with a nonlinear Poincaré inequality which we obtain in this paper, has played an important role in our proof of the above mentioned anisotropic interpolation inequalities.

math.AP↗

arXiv2021

Symmetry of hypersurfaces with ordered mean curvature in one direction

For a connected $n$-dimensional compact smooth hypersurface $M$ without boundary embedded in $\mathbb{R}^{n+1}$, a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a one-directional analog of this result: if every pair of points $(x',a), (x',b)\in M$ with $a<b$ has ordered mean curvature $H(x',b)\leq H(x',a)$, then $M$ is symmetric about some hyperplane $x_{n+1}=c$ under some additional conditions. Their proof was done by the moving plane method and some variations of the Hopf Lemma. We obtain the symmetry of $M$ under some weaker assumptions using a variational argument, giving a positive answer to the conjecture given by Li and Nirenberg.

math.AP↗

arXiv2020

Sharp stability for the interaction energy

This paper is devoted to stability estimates for the interaction energy with strictly radially decreasing interaction potentials, such as the Coulomb and Riesz potentials. For a general density function, we first prove a stability estimate in terms of the $L^1$ asymmetry of the density, extending some previous results by Burchard-Chambers, Frank-Lieb and Fusco-Pratelli for characteristic functions. We also obtain a stability estimate in terms of the 2-Wasserstein distance between the density and its radial decreasing rearrangement. Finally, we consider the special case of Newtonian potential, and address a conjecture by Guo on the stability for the Coulomb energy.

math.AP↗

arXiv2019

Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations

It was proved by Karch and Pilarzyc that Landau solutions are asymptotically stable under any $L^2$-perturbation. In our earlier work with L. Li, we have classified all $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.

math.AP↗

arXiv2019

Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations

We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the critical power being $m = 2$. In the case $m \ge 2$, we show that for any attractive potential the steady state is unique for a fixed mass. In the case $1 < m < 2$, we construct examples of smooth attractive potentials, such that there are infinitely many radially decreasing steady states of the same mass. For the uniqueness proof, we develop a novel interpolation curve between two radially decreasing densities, and the key step is to show that the interaction energy is convex along this curve for any attractive interaction potential, which is of independent interest.

math.AP↗

arXiv2019

Vanishing viscosity limit for homogeneous axisymmetric no-swirl solutions of stationary Navier-Stokes equations

$(-1)$-homogeneous axisymmetric no-swirl solutions of three dimensional incompressible stationary Navier-Stokes equations which are smooth on the unit sphere minus the north and south poles have been classified, %as a four parameter family for each viscosity. In this paper we study the vanishing viscosity limit of sequences of these solutions. As the viscosity tends to zero, some sequences of solutions $C^m_{loc}$ converge to solutions of Euler equations on the sphere minus the poles, while for other sequences of solutions, transition layer behaviors occur. For every latitude circle, there are sequences which $C^m_{loc}$ converge respectively to different solutions of the Euler equations on the spherical caps above and below the latitude circle. We give detailed analysis of these convergence and transition layer behaviors.

math.AP↗

arXiv2019

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. III. Two singularities

All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.

math.AP↗

arXiv2017

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity

We classify all $(-1)-$homogeneous axisymmetric no swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south pole, parameterize them as a two dimensional surface with boundary, and analyze their pressure profiles near the north pole. Then we prove that there is a curve of $(-1)-$homogeneous axisymmetric solutions with nonzero swirl, having the same smoothness property, emanating from every point of the interior and one part of the boundary of the solution surface. Moreover we prove that there is no such curve of solutions for any point on the other part of the boundary. We also establish asymptotic expansions for every (-1)-homogeneous axisymmetric solutions in a neighborhood of the singular point on the unit sphere.

math.AP↗

arXiv2017

Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. II. Classification of axisymmetric no-swirl solutions

We classify all (-1)-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles, parameterizing them as a four dimensional surface with boundary in appropriate function spaces. Then we establish smoothness properties of the solution surface in the four parameters. The smoothness properties will be used in a subsequent paper where we study the existence of (-1)-homogeneous axisymmetric solutions with non-zero swirl on $\mathbb{S}^2\setminus\{S,N\}$, emanating from the four dimensional solution surface.

math.AP↗