arXiv subjects

Changjun Zou

Publications and source records attributed to Changjun Zou.

18 recordsLinked to original sources

arXiv2025

Steady vortex patches on flat torus with a constant background vorticity

We construct a series of vortex patch solutions in a doubly-periodic rectangular domain (flat torus), which is accomplished by studying the contour dynamic equation for patch boundaries. We will illustrate our key idea by discussing the single-layered patches as the most fundamental configuration, and then investigate the general construction for $N$ patches near a point vortex equilibrium. Different with the case of bounded domains in $\mathbb R^2$, a constant background vorticity will arise from the compact nature of flat torus, and the $2$-dimensional translational invariance will bring troubles on determining patch locations. To overcome these two difficulties, we will add additional terms for background vorticity and introduce a centralized condition for location vector. By utilizing the regularity difference of terms in contour dynamic equations, we also obtain the $C^\infty$ regularity and convexity of boundary curves.

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arXiv2024

Regularization for point vortices on $\mathbb S^2$

We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch solutions, whose limit is the well-known von K\'arm\'an point vortex street on $\mathbb S^2$; then we consider the general steady case, where besides a non-localized part induced by the sphere rotation, $j$ positive and $k$ negative patches are located near a nondegenerate critical point of the Kirchhoff--Routh function on $\mathbb S^2$. Our construction is accomplished by Lyapunov--Schmidt reduction argument, where the traveling speed or vortex patch location are used to eliminate the degenerate direction of a linearized operator. We also show that the boundary of each vortex patch is a $C^1$ close curve, which is a perturbation of a small ellipse in the spherical coordinates. As far as we know, this is the first attempt for a regularization of the point-vortex equilibria on $\mathbb S^2$.

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arXiv2024

$C^1$ type regularization for point vortices on $\mathbb S^2$

We construct a series of classic vorticity solutions for incompressible Euler equation on $\mathbb S^2$, which constitute the $C^1$ type regularization for a general traveling point vortex system. The construction is accomplished by applying tangent mapping on $\mathbb S^2$ and Lyapunov--Schmidt reduction argument. Using the fixed-point theorem and a finite dimensional equation on vortex dynamics, we prove that the vortices are located near a nondegenerate critical point of Kirchhoff--Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.

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arXiv2022

Uniqueness and stability of steady vortex rings for 3D incompressible Euler equation

In this paper, we are concerned with the uniqueness and nonlinear stability of vortex rings for the 3D Euler equation. By utilizing Arnold 's variational principle for steady states of Euler equations and concentrated compactness method introduced by P. L. Lions, we first establish a general stability criteria for vortex rings in rearrangement classes, which allows us to reduce the stability analysis of certain vortex rings to the problem of their uniqueness. Subsequently, we prove the uniqueness of a special family of vortex rings with a small cross-section and polynomial type distribution function. These vortex rings correspond to global classical solutions to the 3D Euler equation and have been shown to exist by many celebrate works. The proof is achieved by studying carefully asymptotic behaviors of vortex rings as they tend to a circular filament and applying local Pohozaev identities. Consequently, we provide the first family of nonlinear stable classical vortex ring solutions to the 3D Euler equation.

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arXiv2022

Existence, uniqueness and stability of steady vortex rings of small cross-section

This paper is concerned with steady vortex rings in an ideal fluid of uniform density, which are special global axi-symmetric solutions of the three-dimensional incompressible Euler equation. We systematically establish the existence, uniqueness and nonlinear orbital stability of steady vortex rings of small cross-section for which the potential vorticity is constant throughout the core. The latter two answer a long-standing question since the pioneering work of Fraenkel and Berger \cite{BF1} (Acta Math., 1974). To achieve our goal, we rescale the Stokes stream function of vortex ring by its cross-section radius, and expand it at the well-known Rankine vortex using Taylor's formula, where the estimates for coefficients are obtained by a decomposition according to Green's function and local Pohozaev identities. The main observations are: The stream function is even and has a translational invariance in $z$-direction; the zero point for the $r$-coefficient of linear term in its expansion determines the asymptotic location of vortex ring, which appears as the condition to eliminate the degenerate direction in Lyapunov-Schmidt reduction argument for existence; the non-vanishing condition of the second order $r$-coefficient at foresaid zero point is verified as one of the essential factors for uniqueness, while the negativity means that these vortex rings maximizes the functional composed of kinetic energy and impulse. By applying the Arnol'd's dual variational principle together with the uniqueness result, we are finally able to prove the nonlinear orbital stability of thin vortex rings. This result gives a large class of stable vortex rings supported on topological tori, which is different from Hill's spherical vortex discussed by Choi \cite{Choi20} (Comm. Pure Appl. Math., 2023).

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arXiv2021

On the existence of vortex-wave systems to inviscid gSQG equation

We study the existence of different vortex-wave systems for inviscid gSQG flow, where the total circulation are produced by point vortices and vortices with compact support. To overcome several difficulties caused by the singular formulation and infinite kinetic energy, we introduce a modified reduction method. Several asymptotic properties of the system are also given.

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arXiv2021

Existence of stationary vortex sheets for the 2D Euler equation

We investigate a steady planar flow of an ideal fluid in a (bounded or unbounded) domain $\Omega\subset \mathbb{R}^2$. Let $\kappa_i\not=0$, $i=1,\ldots, m$, be $m$ arbitrary fixed constants. For any given non-degenerate critical point $\mathbf{x}_0=(x_{0,1},\ldots,x_{0,m})$ of the Kirchhoff-Routh function defined on $\Omega^m$ corresponding to $(\kappa_1,\ldots, \kappa_m)$, we construct a family of stationary planar flows with vortex sheets that have large vorticity amplitude and are perturbations of small circles centered near $x_i$, $i=1,\ldots,m$. The proof is accomplished via the implicit function theorem with suitable choice of function spaces. This seems to be the first nontrivial result on the existence of stationary vortex sheets in domains.

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arXiv2021

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.

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arXiv2021

K\'arm\'an vortex street for the generalized surface quasi-geostrophic equation

We are concerned with the existence of periodic travelling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation(including incompressible Euler equation), known as von K\'arm\'an vortex street. These solutions are of $C^1$ type, and are obtained by studying a semilinear problem on an infinite strip whose width equals to the period. By a variational characterization of solutions, we also show the relationship between vortex size, travelling speed and street structure. In particular, the vortices with positive and negative intensity have equal or unequal scaling size in our construction, which constitutes the regularization for K\'arm\'an point vortex street.

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arXiv2021

On the global classical solutions for the generalized SQG equation

In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions for to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and travelling-wave solutions.

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arXiv2021

Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations

By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.

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arXiv2021

Global solutions for the generalized SQG equation and rearrangements

In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with $N$-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating $N$ point vortices and counter-rotating pairs. Some other quantitative properties are also established.

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arXiv2021

Existence and regularity of co-rotating and travelling global solutions for the generalized SQG equation

By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu \cite{HM} to $\alpha\in[1,2)$. Moreover, we prove the $C^\infty$ regularity of vortices boundary, and show the convexity of each vortices component.

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arXiv2021

Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation

In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible functions. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Using these facts, we further prove the orbital stability of the circular vortex pairs for the gSQG equation.

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arXiv2020

Multi-soliton dynamics in the nonlinear Schr\"{o}dinger equation

In this paper, we study the Cauchy problem of the nonlinear Schr\"{o}dinger equation with a nontrival potential $V_\varepsilon(x)$. In particular, we consider the case where the initial data is close to a superposition of $k$ solitons with prescribed phase and location, and investigate the evolution of the Schr\"{o}dinger system. We prove that over a large time interval with the maximum time tending to infinity, all $k$ solitons will maintain the shape, and the solitons dynamics can be regarded as an approximation of $k$ particles moving in $\mathbb{R}^N$ with their accelerations dominated by $\nabla V_\varepsilon$, provided the barycenters of these solitons do not coincide.

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arXiv2020

Local uniqueness of vortices for 2D steady Euler flow in a bounded domain

We study the 2D Euler equation in a bounded simply-connected domain, and establish the local uniqueness of flow whose stream function $\psi_\varepsilon$ satisfies \begin{equation*} \begin{cases} -\varepsilon^2\Delta \psi_\varepsilon=\sum\limits_{i=1}^k \mathbf1_{B_\delta(z_{0,i})}(\psi_\varepsilon-\mu_{\varepsilon,i})_+^\gamma,\ \ \ & \text{in} \ \Omega, \psi_\varepsilon=0,\ \ \ & \text{on} \ \Omega, \end{cases} \end{equation*} with $\varepsilon\to 0^+$ the scale parameter of vortices, $\gamma\in(0,\infty)$, $\Omega\subset \mathbb R^2$ a bounded simply connected Lipschitz domain, $z_{0,i}\in\Omega$ the limiting location of $i^{\text{th}}$ vortex, and $\mu_{\varepsilon,i}$ the flux constants unprescribed. Our proof is achieved by a detailed description of asymptotic behavior for $\psi_\varepsilon$ and Pohozaev identity technique. For $k=1$, we prove the nonlinear stability of corresponding vorticity in $L^p$ norm, provided $z_{0,1}$ is a non-degenerate minimum point of Robin function. This stability result can be generalized to the case $k\ge 2$, and $(z_{0,1},\cdots,z_{0,k})\in \Omega^k$ being a non-degenerate minimum point of the Kirchhoff-Routh function.

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arXiv2019

Desingularization of steady vortex of perturbation type in the lake equations

In this paper, we study the desingularization of steady lake model of perturbation type with general nonlinearity f. Using the modified vorticity method, we construct a family of steady solutions with vanishing circulation, which constitute a desingularization of a singular vortex. The localization of the singular vortex is determined only by the vanishing rate of the circulation. Some qualitative and asymptotic properties are also established.

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arXiv2019

On desingularization of steady vortex in the lake equations

We constructed a family of steady vortex solutions for the lake equations with general vorticity function, which constitute a desingularization of a singular vortex. The precise localization of the asymptotic singular vortex is shown to be the deepest position of the lake. We also study global nonlinear stability for these solutions. Some qualitative and asymptotic properties are also established.

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