arXiv subjects

Guolin Qin

Publications and source records attributed to Guolin Qin.

At least 19 recordsLinked to original sources

arXiv2026

Planar Gross--Pitaevskii traveling waves at every subsonic speed

For every subsonic speed $c\in(0,\sqrt2)$, we prove the existence of a finite-energy traveling wave for the planar Gross--Pitaevskii equation. This resolves the longstanding problem of the existence of prescribed-speed traveling-wave solutions in two dimensions, explicitly stated as open by Mari\c{s} (Ann. of Math., 2013) and Bellazzini and Ruiz (Amer. J. Math., 2023). The proof relies essentially on the energy estimate \begin{equation*} E(\psi)\le C_J\bigl(I_c(\psi)+\ind(\psi)\bigr), \qquad c\in J, \end{equation*} where $E$ is the energy, $I_c$ the action at speed $c$, $\ind$ the real Morse index, $J$ is any compact interval contained in $(0,\sqrt2)$, and $C_J$ is a positive constant depending only on $J$. We also prove finite-bubble compactness, including splitting of the energy, action, potential energy, and momentum, and attainment of the action among nonconstant waves of Morse index at most one.

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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

Optimal Rigidity Results for the $k$-Hessian Equation of Lane--Emden Type

In this paper, we establish optimal Liouville theorems and classification results for the $k$-Hessian Lane--Emden equation \[ \sigma_k(-D^2u)=u^p\quad\text{in }\R^n,\qquad -D^2u\in\overline{\Gamma_k},\qquad u\geq 0, \] where $2\leq k<\frac{n}{2}$ and $p>0$. Let $p_- = \frac{nk}{n-2k}$ and the critical Hessian--Sobolev exponent $p_* = \frac{(n+2)k}{n-2k}$. Phuc and Verbitsky proved nonexistence of positive solutions for $k 2k$, without any additional assumption. For the limiting case $n=2k$, we classify finite-mass solutions to the $\frac{n}{2}$-Hessian Liouville equation under a proper asymptotic condition $u(x)\rightarrow-\infty$ as $|x|\rightarrow\infty$. In particular, we provide the fully nonlinear counterparts of the classical Liouville and classification theorems of Gidas--Spruck, Gidas--Ni--Nirenberg, and Caffarelli--Gidas--Spruck.

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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 M\"obius--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.

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arXiv2026

A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin--Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

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arXiv2026

Flexibility and rigidity for the Couette flow in the infinite channel

We investigate the existence of stationary and traveling wave solutions to the 2D Euler equations near the Couette flow in the infinite channel $\mathbb{R} \times [-1,1]$. For Sobolev spaces $W^{s,p}$ or H\"older spaces $C^s$, we identify the index $s= 1+ \frac1p $ as the vorticity regularity threshold separating flexibility from rigidity. Specifically, for any $s<1+ \frac1p$ we prove the existence of $C^\infty$ smooth, compactly supported steady states and traveling waves arbitrarily close to the Couette flow in all $W^{s,p}$ and $C^{1-}$. Conversely, we establish the non-existence of such relative equilibria in $ W^{s,p}$ with $s>1+ \frac1p$ or $C^{1+}$. A notable feature of the variational construction is that these flexible solutions belong to every Gevrey class strictly below the analytic threshold.

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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

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

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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arXiv2024

On concentrated vortices of 3D incompressible Euler equations under helical symmetry: with swirl

In this paper, we consider the existence of concentrated helical vortices of 3D incompressible Euler equations with swirl. First, without the assumption of the orthogonality condition, we derive a 2D vorticity-stream formulation of 3D incompressible Euler equations under helical symmetry. Then based on this system, we deduce a non-autonomous second order semilinear elliptic equations in divergence form, whose solutions correspond to traveling-rotating invariant helical vortices with non-zero helical swirl. Finally, by using Arnold's variational method, that is, finding maximizers of a properly defined energy functional over a certain function space and proving the asymptotic behavior of maximizers, we construct families of concentrated traveling-rotating helical vortices of 3D incompressible Euler equations with non-zero helical swirl in infinite cylinders. As parameter $ \varepsilon\to0 $, the associated vorticity fields tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow.

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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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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).

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arXiv2023

Method of scaling spheres: Liouville theorems in inner or outer (unbounded or bounded) generalized radially convex domains, blowing-up analysis on domains with not necessarily $C^1$ boundary and other applications

In this paper, we aim to introduce the method of scaling spheres (MSS) as a unified approach to Liouville theorems on general domains in $\mathbb R^n$, and apply it to establish Liouville theorems on arbitrary unbounded or bounded MSS applicable domains for general ($\leq n$-th order) PDEs and integral equations without translation invariance or with singularities. The set of MSS applicable domains includes any unbounded or bounded generalized radially convex domains and any complementary sets of their closures, which is invariant under Kelvin transforms and is the maximal collection of simply connected domains such that the MSS works. For instance, $ \mathbb R^n$, $\mathbb R^n_+$, balls, cone-like domains, convex domains, star-shaped domains and all the complements of their closures are MSS applicable domains. MSS applicable domains is to the MSS what convex domains is to the method of moving planes. As applications, we derive a priori estimates and existence of solutions from the boundary Hölder estimates for Dirichlet or Navier problems of Lane-Emden equations by applying the blowing-up argument on domains with blowing-up cone boundary (BCB domains for short). After the blowing-up procedure, the BCB domains allow the limiting shape of the domain to be a cone (half space is a cone). While the classical blowing-up techniques in previous papers work on $C^1$-smooth domains, we are able to apply blowing-up analysis on more general BCB domains on which the boundary Hölder estimates hold (can be guaranteed by uniform exterior cone property etc). Since there are no smoothness conditions on the boundary of MSS applicable domain and BCB domain, our results clearly reveal that the existence and nonexistence of solutions mainly rely on topology (not smoothness) of the domain.

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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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arXiv2022

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity

In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{2}$: \begin{equation*}\\\begin{cases} (-Δ)^{\frac{1}{2}}u(x)=e^{pv(x)}, \qquad x\in\mathbb{R}^{2}, \\ -Δv(x)=u^{4}(x), \qquad x\in\mathbb{R}^{2}, \end{cases}\end{equation*} where $p\in(0,+\infty)$, $u\geq 0$ and satisfies the finite total curvature condition $\int_{\mathbb{R}^{2}}u^{4}(x)\mathrm{d}x<+\infty$. In order to show integral representation formula and crucial asymptotic property for $v$, we derive and use an $\exp^{L}+L\ln L$ inequality, which is itself of independent interest. When $p=\frac{3}{2}$, the system is closely related to single conformally invariant equations $(-Δ)^{\frac{1}{2}}u=u^{3}$ and $-Δv=e^{2v}$ on $\mathbb{R}^{2}$, which have been quite extensively studied (cf. \cite{BF,C,CY,CL,CLL,CLZ} etc). We also derive classification results for nonnegative solutions to conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in $\mathbb{R}^{3}$. Extensions to mixed order conformally invariant systems in $\mathbb{R}^{n}$ with general dimensions $n\geq3$ are also included.

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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.

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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

Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

In this paper, we establish various maximum principles and develop the method of moving planes and the sliding method (on general unbounded domains) for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or $\mathbb{R}^{n}$. In particular, the uniformly elliptic nonlocal Monge-Ampère operator introduced by Caffarelli and Charro in \cite{CC} is a typical example of the uniformly elliptic nonlocal Bellman operator.

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