arXiv subjects

Weicheng Zhan

Publications and source records attributed to Weicheng Zhan.

At least 19 recordsLinked to original sources

arXiv2026

A Variational Family of Traveling Vortex Dipoles: Transition, Uniqueness, and Stability

We provide a unified description of a stable variational family of traveling vortex dipoles for the two-dimensional incompressible Euler equations, connecting the classical Lamb dipole to asymptotically radial dipoles in the large-impulse regime. We show that this family undergoes a sharp transition from the Lamb dipole, where the mass constraint is inactive, to non-explicit traveling waves for which the mass constraint becomes active, and determine the critical value exactly. We further prove that the variational solutions are unique up to translation both near the Lamb dipole and in the large-impulse regime. As a consequence, the corresponding individual traveling waves are orbitally stable. Our results provide a unified picture of these two distinct asymptotic regimes within a single variational family of traveling vortex dipoles.

math.AP↗

arXiv2026

Proof of the planar Khavinson-Shapiro conjecture

Let $Ω\subset\mathbb{R}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson--Shapiro conjecture in this setting: if every polynomial on $\mathbb{R}^2$ agrees on $\partialΩ$ with a harmonic polynomial, then $\partialΩ$ is an ellipse and $Ω$ is its bounded interior.

math.AP↗

arXiv2026

Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine Holes

For every prescribed integer $N\ge2$, we construct uniformly rotating unit-vorticity vortex patches $ω=\mathbf{1}_D$ for the planar Euler equation such that $D$ is connected and $\mathbb{R}^2\backslash D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes, rather than opposite-sign vortex patches or regions carrying a second nonzero vorticity level. The angular velocities lie in the rigidity-compatible interval $(0,1/2)$, and the domains converge in measure to the Rankine disk as the holes collapse. The construction starts from a fixed co-rotating polygonal configuration of $N$ unit-vorticity vortex patches and removes a shrinking spatial copy of that configuration from the Rankine disk. An exact complement identity solves all inner-boundary equations before the outer circle is perturbed. The remaining defect is generated by the $N$-th exterior multipole and has size $\varepsilon^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon^{2N}$. Separate renormalization of the outer and inner equations produces a limiting affine system with a lower-triangular derivative. Its diagonal blocks are the nonresonant Rankine operator and the angular-velocity-augmented linearization of the fixed seed configuration. We also determine the first corrections to the outer boundary, the hole boundaries, and the angular velocity.

math.AP↗

arXiv2026

Smooth Failure of Boundary Unique Continuation for Harmonic Functions

For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a Möbius--Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15--16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.

math.AP↗

arXiv2025

Remarks on radial symmetry of stationary and uniformly-rotating solutions for the 2D Euler equation

We prove that any uniformly rotating solution of the 2D incompressible Euler equation with compactly supported vorticity $ω$ must be radially symmetric whenever its angular velocity satisfies $Ω\in (-\infty,\inf ω/ 2] \cup \, [ \sup ω/ 2, +\infty )$, in both the patch and smooth settings. This result extends the rigidity theorems established in \cite{Gom2021MR4312192} (\textit{Duke Math. J.},170(13):2957-3038, 2021), which were confined to the case of non-positive angular velocities and non-negative vorticity. Moreover, our results do not impose any regularity conditions on the patch beyond requiring that its boundary consists of Jordan curves, thereby refining the previous result to encompass irregular vortex patches.

math.AP↗

arXiv2025

Symmetry in Serrin-type overdetermined problems

This paper investigates the geometric constraints imposed on a domain by overdetermined problems for partial differential equations. Serrin's symmetry results are extended to overdetermined problems with potentially degenerate ellipticity in nonsmooth bounded domains. Furthermore, analogous symmetry results are established for ring-shaped domains. The proof relies on continuous Steiner symmetrization, along with a carefully constructed approximation argument.

math.AP↗

arXiv2024

Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler equation in the unit disc, both in the smooth setting and the patch setting. In the patch setting, we prove that every uniformly rotating patch with angular velocity $Ω\le 0$ or $Ω\ge 1/2$ must be radial, where both bounds are sharp. The conclusion holds under the assumption that the rotating patch considered is disconnected, with its boundaries consisting of several Jordan curves. We also show that every uniformly rotating smooth solution $ω_0$ must be radially symmetric if its angular velocity $Ω\le \inf ω_0/2$ or $Ω\ge \sup ω_0/2$. The proof is based on the symmetry properties of non-negative solutions to elliptic problems. A newly tailored approach is developed to address the symmetries of non-negative solutions to piecewise coupled semi-linear elliptic equations.

math.AP↗

arXiv2024

On the rigidity of the 2D incompressible Euler equations

We consider rigidity properties of steady Euler flows in two-dimensional bounded domains. We prove that steady Euler flows in a disk with exactly one interior stagnation point and tangential boundary conditions must be circular flows, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. Moreover, for steady Euler flows on annuli with tangential boundary conditions, we prove that they must be circular flows provided there is no stagnation point inside, which answers another open problem proposed by F. Hamel and N. Nadirashvili in the same paper. We secondly show that the no-slip boundary conditions would result in absolute rigidity in the sense that except for the disks (\emph{resp}. annuli), there is no other smooth simply (\emph{resp}. doubly) connected bounded domain on which there exists a steady flow with only one (\emph{resp}. no) interior stagnation point and no-slip boundary conditions, and if present on the other hand, the flow must be circular. The arguments are based on the geometry of streamlines and 'local' symmetry properties for the non-negative solutions of semi-linear elliptic problems.

math.AP↗

arXiv2024

Free boundary problems for the two-dimensional Euler equations in exterior domains

In this paper we present some classification results for the steady Euler equations in two-dimensional exterior domains with free boundaries. We prove that, in an exterior domain, if a steady Euler flow devoid of interior stagnation points adheres to slip boundary conditions and maintains a constant norm on the boundary, along with certain additional conditions at infinity, then the domain is the complement of a disk, and the flow is circular, namely the streamlines are concentric circles. Additionally, we establish that in the entire plane, if all the stagnation points of a steady Euler flow coincidentally form a disk, then, under certain additional reasonable conditions near the stagnation points and at infinity, the flow must be circular. The proof is based on a refinement of the method of moving planes.

math.AP↗

arXiv2023

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.

math.AP↗

arXiv2023

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).

math.AP↗

arXiv2023

A rigidity result for the Euler equations in an annulus

We are concerned with rigidity properties of steady Euler flows in two-dimensional bounded annuli. We prove that in an annulus, a steady flow with no interior stagnation point and tangential boundary conditions is a circular flow, which addresses an open question proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. The proof is based on the study of the geometric properties of the streamlines of the flow and on `local' symmetry properties for the non-negative solutions of semi-linear elliptic equations with a continuous nonlinearity.

math.AP↗

arXiv2023

A Liouville theorem for the Euler equations in a disk

We present a symmetry result regarding stationary solutions of the 2D Euler equations in a disk. We prove that in a disk, a steady flow with only one stagnation point and tangential boundary conditions is a circular flow, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in [J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. The key ingredient of the proof is to use `local' symmetry properties for the non-negative solutions of semi-linear elliptic equations with a continuous nonlinearity in a ball, which can be established by a rearrangement technique called continuous Steiner symmetrization.

math.AP↗

arXiv2022

Existence and Stability of the Lamb Dipoles for the Quasi-Geostrophic Shallow-Water Equations

In this paper, we prove the nonlinear orbital stability of vortex dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are explicit travelling wave solutions to the QGSW equations, which are analogues of the classical circular vortex of Lamb and Chaplygin for the steady planar Euler equations. We establish a variational characterization of these vortex poles, which provides a basis for the stability result.

math.AP↗

arXiv2021

Kármán 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 Kármán 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 Kármán point vortex street.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗