arXiv subjects

Daomin Cao

Publications and source records attributed to Daomin Cao.

At least 19 recordsLinked to original sources

arXiv2026

Global persistence of nearly radial concentrated vortices in a bounded domain

In this paper, we study the evolution of nearly radial concentrated vortices for the incompressible Euler equation in a bounded planar domain. We prove that if a single vortex is initially concentrated near a strict local minimum point of the Robin function of the domain and is close, up to translation, to its symmetric decreasing rearrangement, then both its shape and location remain uniformly controlled for all time. We also establish an analogous result for a pair of opposite-sign vortices near a strict local minimum point of the corresponding Kirchhoff--Routh function. No symmetry is imposed on the domain or the initial data, and the initial data need not be close to any steady state. To prove these results, we develop a new Lyapunov mechanism for the evolution of vorticity governed by the Euler equation starting from such initial data, providing quantitative control of both vortex shape and location. Specifically, we combine kinetic-energy conservation and vorticity equimeasurability with several fixed-time estimates to obtain a conditional estimate, and then use a set-valued first-exit argument to propagate it globally in time.

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arXiv2026

Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows

We consider the zero-homogeneous reduction of the two-dimensional Euler equation in the 3-fold symmetric case $m=3$, which is left unsolved in the $m\geq4$ relaxation theory of Said, Elgindi, and Murray \cite{EMS}. We prove that every nonconstant $W^{1,p}$ solution satisfies $\|g_\theta(t)\|_{L^p}\to\infty$ as $t\to\pm\infty$ for $1<p\leq\infty$. For $p=1$, the total variation is conserved, but the $L\log L$ modular tends to infinity whenever it is initially finite. If $D_\theta g_0$ is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in $W^{\alpha,r}$ for $\alpha r<1$. The structural assumption on $D_\theta g_0$ is automatic for $C^1$ data. Moreover, every weak $L^2$ infinite-time limit generates a complete $L^2$-precompact orbit.

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arXiv2026

Growth of vorticity gradient for the Euler equation on the sphere

We prove that for solutions of the Euler equation on the sphere, the vorticity gradient can grow at most double-exponentially in time, and we show that this upper bound is sharp by constructing explicit solutions with odd symmetry that exhibit double-exponential growth in the hemisphere. We also extend the results to the case of a rotating sphere. This seems to be the first result on the growth of the vorticity gradient for ideal fluids on a compact manifold with non-trivial geometry.

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arXiv2025

Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations

In this article, we consider clustered solutions to a semilinear elliptic equation in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in \Omega,\\ u=0,\ \ &x\in\partial \Omega \end{cases} \end{equation*} for small values of $ \varepsilon $. Using Green's function of the elliptic operator $ -\text{div}(K(x)\nabla) $ and finite-dimensional reduction method, we prove that there exist clustered solutions with cluster point $ 0 $ and cluster distance $ |\ln\varepsilon| ^{-\frac{1}{2}} $ whose small-structure is governed by some functional $ H_N $ determined by $ K $ and $ q $. As an application, we prove the existence of traveling-rotating helical vorticity fields to 3D incompressible Euler equations in infinite cylinders, whose support sets consist of helical tubes with small cross-section of radius $ \varepsilon $ and arbitrary circulation $ \kappa $ and concentrates near ``$ 2N $'' and ``$ 2N+1 $'' type of co-rotating helical solutions of nearly parallel vortex filaments model as $ \varepsilon\to0 $, which justifies the result in Klein, Majda and Damodaran [1995, JFM] and generalizes results in Guerra and Musso [arxiv: 2502.01470]. Several kinds of solutions such as ``2 asymmetric'', ``$ 2\times2 $ asymmetric'' and ``$ 2\times2+1 $ asymmetric'' type of co-rotating helical filaments are also considered.

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arXiv2025

Linear Growth of the Vorticity Maximum for Axisymmetric Euler Flows Without Swirl

We study long-time vortex stretching for three-dimensional axisymmetric Euler flows without swirl in the anti-parallel class associated with the head-on collision of two coaxial vortex rings. This geometry motivated Childress's $t^{4/3}$ conjecture for the vorticity maximum in the full axisymmetric no-swirl class [S.~Childress, \emph{Physica D} \textbf{237} (2008), 1921--1925]. For unit-strength relative-vorticity patches in this class, we prove that the outer radius, which is exactly the vorticity maximum, reaches the linear scale on an arbitrarily large fixed proportion of every sufficiently large dyadic interval: for every $0<\eta<1$, there exist $c_\eta>0$ and $T_\eta>1$ such that \[ \left| \left\{t\in[T,2T]: \mathcal R_\omega(t) =\|\boldsymbol\Omega(t)\|_{L^\infty(\mathbb R^3)} \ge c_\eta t \right\} \right|\ge(1-\eta)T \qquad(T\ge T_\eta). \] The same estimate for $\mathcal R_\omega(t)$ holds for all data considered below. For every nontrivial compactly supported initial datum in this class that is odd in $z$ and non-positive for $z>0$, we also prove \[ \lim_{t\to\infty} \frac{P(t)[\log(2+t)]^{5/2}}{(1+t)^{3/2}} =+\infty. \] To the best of our knowledge, this is the first radial-moment lower bound with exponent greater than one. For unit-strength patches, the same moment bound also yields the full-time estimate \[ \lim_{t\to\infty} \frac{\|\boldsymbol\Omega(t)\|_{L^\infty(\mathbb R^3)} [\log(2+t)]^{5/4}}{(1+t)^{3/4}} =+\infty. \] For general data, we further obtain a quantitative Eulerian form of simultaneous radial escape and collision. The proof uses two monotone mixed moments, a compactly supported multiplier, and an exterior $L^2$ estimate for the velocity generated by interior vorticity.

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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.

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arXiv2025

Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations

In this paper, we investigate the time evolution of helical vortices without swirl for the incompressible Euler equations in $\mathbb R^3$ under general initial assumptions. Assume the initial helical vorticity is sharply concentrated in $N$ distinct $\ep$-neighborhoods, whose mutual distances vanish as $O(1/|\ln \ep|)$, and each vortex core possesses vorticity mass of order $1/|\ln \ep|^{1+b}$ for an arbitrary fixed $b\in\mathbb R$. We prove that as $\ep\to 0$, the motion of these helical vortices converges uniformly to a dynamical system derived herein over a time interval of order $1/|\ln\varepsilon|^{1-b}$. In the particular case $b=-1$, our results establish the evolution counterpart for interacting vortex helices constructed in [I. Guerra, M. Musso, Ann. Inst. H. Poincar\'e C Anal. Non Lin\'aire, 2025]. Notably, for two interacting helical vortices with initial mutual distance $ \rho_0/|\ln \ep|$, by choosing $\rho_0$ sufficiently small, our analysis extends to timescales covering multiple periods. This result provides the first mathematical justification for the numerically observed phenomenon termed ``leapfrogging of Kelvin waves" reported in [N. Hietala et al., Phys. Rev. Fluids, 2016].

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arXiv2025

Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^{1,p}$-critical quasi-linear nonlocal elliptic equations with Hardy potential

In this paper, we mainly consider nonnegative weak solutions $u\in D^{1,p}(\R^{N})$ to the doubly $D^{1,p}(\R^{N})$-critical nonlocal quasi-linear Schr\"{o}dinger-Hartree equation: \begin{align*} -\Delta_p u- \mu \frac{u^{p-1}}{|x|^p}=\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $N\geq3$, $0\leq\mu< \bar{\mu}:=\left( (N-p)/p \right)^p$ and $1 0$, due to appearance of the Hardy potential, the equation has singularity at $0\in\mathbb{R}^{N}$ and hence is not translation invariant, so sharp asymptotic estimates near the origin must be involved. First, we establish regularity and the sharp estimates on asymptotic behaviors near the origin and the infinity for any positive solution $u\in D^{1,p}(\R^{N})$ (and $|\nabla u|$) to more general equation $-\triangle_p u - \mu \frac{1}{|x|^p}u^{p-1}=V(x)\frac{1}{|x|^s}u^{p-1}$ with $N\geq2$, $0\leq\mu< \bar{\mu}$, $1<p<N$, $0\leq s < p$ and $0\leq V(x)\in L^\frac{N}{p-s}(\R^N)$. Then, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions in $D^{1,p}(\R^{N})$ are radially symmetric and strictly radially decreasing about the origin $0\in\mathbb{R}^{N}$. The sharp asymptotic estimates and radial symmetry for more general weighted doubly $D^{1,p}$-critical nonlocal quasi-linear equations were also derived. Our results extend the results in \cite{DLL} from the special case $\mu=0$ to general cases $0\leq\mu<\bar{\mu}$.

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arXiv2024

Desingularization of vortices for the incompressible Euler equation on a sphere

In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations.

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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.

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arXiv2024

On Arnold-type stability theorems for the Euler equation on a sphere

In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere. Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stable in the Lyapunov sense. In particular, our theorems apply to degree-2 Rossby-Haurwitz waves. These results are achieved via a variational approach, with the key ingredient being to show that the flows under consideration satisfy the conditions of two Burton-type stability criteria which are established in this paper. As byproducts, we obtain some sharp rigidity results for solutions of semilinear elliptic equations on a sphere.

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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.

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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.

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arXiv2024

Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form

We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ \Omega\subseteq\mathbb{R}^n (n\geq 2) $ with zero Dirichlet boundary condition, where $ K $ is a smooth positively definite matrix-valued function on $ \Omega $. We obtain a high-order asymptotic expansion of $ G_K(x, y) $, which defines uniquely a regular part $ H_K(x, y) $. Moreover, we prove that the associated Robin's function $ R_K(x) = H_K(x, x) $ is smooth in $ \Omega $, despite the regular part $ H_K\notin C^1(\Omega\times\Omega) $ in general.

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arXiv2023

Clustered helical vortices for 3D incompressible Euler equation in infinite cylinders

In this article, we first consider solutions to a semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\\ u=0,\ \ &x\in\partial Ω\end{cases} \end{equation*} for small values of $ \varepsilon $. We prove that there exists a family of clustered solutions which have arbitrary many bubbles and collapse into given maximum points of $ q^2\sqrt{\det K} $ as $ \varepsilon\to0. $ Then as an application, we construct clustered traveling-rotating helical vortex solutions to Euler equations in infinite cylinders, such that the support set of corresponding vortices consists of several helical tubes concentrating near a single helix.

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arXiv2023

Construction of multi-bubble blow-up solutions to the $L^2$-critical half-wave equation

This paper concerns the bubbling phenomena for the $L^2$-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in \cite{K-L-R} for the single-bubble case, and explores the localization techniques in \cite{CSZ21,RSZ21} for bubbling solutions to nonlinear Schrödinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control strong interactions, as well as nonlocal effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders $|t|^{2+}$ and $|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.

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arXiv2023

Stability of degree-2 Rossby-Haurwitz waves

Rossby-Haurwitz (RH) waves are important explicit solutions of the incompressible Euler equation on a two-dimensional rotating sphere. In this paper, we prove the orbital stability of degree-2 RH waves, which confirms a conjecture proposed by A. Constantin and P. Germain in [Arch. Ration. Mech. Anal. 245, 587-644, 2022]. The proofs are based on a variational approach, with the main challenge being to establish suitable variational characterizations for the solutions under consideration. In this process, the set of rearrangements of a fixed function plays a vital role. We also apply our approach to the stability analysis of degree-1 RH waves, Arnold-type flows, and zonal flows with monotone absolute vorticity.

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arXiv2023

Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.

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