arXiv subjects

Junhong Fan

Publications and source records attributed to Junhong Fan.

4 recordsLinked to original sources

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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].

math.AP↗