arXiv subjects

Boquan Fan

Publications and source records attributed to Boquan Fan.

5 recordsLinked to original sources

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 $\omega$ must be radially symmetric whenever its angular velocity satisfies $\Omega \in (-\infty,\inf \omega / 2] \cup \, [ \sup \omega / 2, +\infty )$, 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↗

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 $\Omega\le 0$ or $\Omega \ge 1/2$ 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 $\omega_0$ must be radially symmetric if its angular velocity $\Omega\le \inf \omega_0/2$ or $\Omega\ge \sup \omega_0/2$. 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

Helical kelvin waves for the 3D Euler equation

Helical Kelvin waves were conjectured to exist for the 3D Euler equations in Lucas and Dritschel \cite{LucDri} (as well as in \cite{Chu}) by studying dispersion relation for infinitesimal linear perturbations of a circular helically symmetric vortex patch. This paper aims to rigorously establish the existence of these $m$-fold symmetric helical Kelvin waves, in both simply and doubly connected cases, for the 3D Euler equations. The construction is based on linearization of contour dynamics equations and bifurcation theory. Our results rigorously verify the prediction in aforementioned papers and extend $m$-waves of Kelvin from the 2D Euler equations to the 3D helically symmetric Euler equations.

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↗

arXiv2024

Symmetry of uniformly rotating solutions for the vortex-wave system

In this paper, we study the radial symmetry properties of stationary and uniformly rotating solutions of the vortex-wave system introduced by Marchioro and Pulvirenti \cite{Mar1}. We show that every uniformly rotating patch $\left(D,x_1,x_2,..,x_k\right)$ with angular velocity $\Omega\leq 0$ must be radial with respect to the only point vortex $x_1$, implying that $k=1$. In other words, the background vorticity consists of finite nested annulus and the point vortex is located at the center of these annulus. In contrast to the case where the angular velocity is non-positive, we prove that there exists a family of uniformly rotating patch $(D^n, x_1^n)_n$ solutions, which are associated with a sequence of positive angular velocities $\{\Omega_n\}$ and are not annular. Furthermore, we find that the set of bifurcating angular velocities $\{\Omega_n\}$ is dense in the interval $(0,+\infty)$, a novel feature that distinguishes this behavior from that observed in the classical Euler equation and gSQG equation.

math.AP↗