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In-Jee Jeong

Publications and source records attributed to In-Jee Jeong.

At least 19 recordsLinked to original sources
arXiv2026

Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}(\mathbb{R}^{d})$ for $d = 2, 3$. In two dimensions, we prove global propagation of the $W^{2,1}(\mathbb{R}^{2})$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}(\mathbb{R}^{3})$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d/p,p}(\mathbb{R}^{d})$ for all $1 < p < \infty$, which were based on axisymmetric flows without swirl when $d = 3$.

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arXiv2026

Global dynamics of a single vortex ring

We study the global-in-time dynamics of vortex rings for the three-dimensional incompressible Euler equations, under the assumption of axisymmetric flows without swirl. For a broad class of initial data sharing only the macroscopic invariants with a thin vortex ring, we prove that the vorticity remains sharply concentrated and propagates along the symmetry axis with leading-order speed given by the Kelvin--Hicks formula, providing the first global-in-time validation of the vortex filament conjecture for a single vortex ring arising from generic initial data. We further identify a universal filamentation mechanism driven by the competition between rapid core translation and slower local induction. This mechanism gives linear-in-time stretching of the vortex support under very general assumptions on the data, yielding dynamical instability of any thin vortex ring configurations in the $W^{2,\infty}$ norm.

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arXiv2025

Vortex atmospheres of traveling vortices: rigorous definition, existence, and topological classification

In incompressible and inviscid fluids, the vortex atmosphere refers to the collection of fluid particles outside the support of a traveling vortex that are nevertheless carried along with it. This phenomenon has been recognized since the nineteenth century, e.g., in the classical works of O. Reynolds [Nature, 1876] and O. Lodge [Lond. Edinb. Dubl. Phil. Mag., 1885], yet rigorous mathematical definitions and proofs have remained largely undeveloped, with most subsequent studies relying on thin-core approximations or asymptotic analyses. In this paper, we give a rigorous definition of a vortex atmosphere and establish its existence and uniqueness. We further compare the planar atmosphere surrounding a 2D vortex dipole with the axisymmetric atmosphere surrounding a 3D vortex ring. In particular, we emphasize and prove the topological distinctions observed by W. Hicks [Lond. Edinb. Dubl. Phil. Mag., 1919]: under natural assumptions, every 2D dipole with its atmosphere forms an oval-shaped region, whereas for 3D rings, both spheroidal and toroidal configurations may occur. Our proof is based on showing that each atmosphere can be characterized precisely as a specific superlevel set of its corresponding stream function.

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arXiv2025

Homogeneous steady states for the generalized surface quasi-geostrophic equations

We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $\psi$ and advected scalar $\omega$ are of the form \begin{align*} \psi=\frac{w(\theta)}{r^{\beta}},\quad \omega=\frac{g(\theta)}{r^{\beta+2s}}, \end{align*} in polar coordinates $(r,\theta)$ with parameter $\beta\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<\beta<-2s$ and $0<\beta<m_0+2$ ($1/2-s<\beta< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq \beta\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions.

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arXiv2025

Low regularity Sobolev well-posedness for Vlasov--Poisson

We consider the Vlasov--Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ having compact support in $v$. In particular, data not belonging to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for large $p$ are allowed.

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arXiv2025

Stability of Lamb dipoles for odd-symmetric and non-negative initial disturbances without the finite mass condition

In this paper, we consider the stability of the Lamb dipole solution of the two-dimensional Euler equations in $\mathbb{R}^{2}$ and question under which initial disturbance the Lamb dipole is stable, motivated by experimental work on the formation of a large vortex dipole in two-dimensional turbulence. We assume (O) odd symmetry for the $x_2$-variable and (N) non-negativity in the upper half plane for the initial disturbance of vorticity, and establish the stability theorem of the Lamb dipole without assuming (F) finite mass condition. The proof is based on a new variational characterization of the Lamb dipole using an improved energy inequality.

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arXiv2025

Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in $\mathbb{R}^{3}$. Namely, for any $s\in (0,3/2)$ and $\varepsilon >0$, we construct a divergence-free initial vorticity $\omega_0$ defined in $\mathbb{R}^{3}$ satisfying $\| \omega_0 \|_{H^s}\leq \varepsilon$, as well as $T>0$, $c>0$ and a corresponding local-in-time solution $\omega$ such that, for each $t\in [0,T]$, $\omega (\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$ and $ \omega (\cdot ,t ) \not \in {H^\beta }$ for any $\beta > \frac{s-ct}{1+ct} $. Moreover, $\omega$ is unique among all solutions with initial condition $\omega_0$ which are locally $C^2$ and belong to $C([0,T];L^p )$ for any $p>3 $.

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arXiv2025

Stability for multiple Lamb dipoles

In the class of nonnegative vorticities on the half-plane, we establish the Lyapunov stability of finite sums of Lamb dipoles under the initial assumptions that the dipoles are sufficiently separated and that the faster dipoles are positioned to the right of the slower ones. Our approach combines sharp energy estimates near the Lamb dipoles with a Lagrangian bootstrapping scheme, enabling us to quantify the exchanges of circulation, enstrophy, impulse, and energy between various parts of the solution. The strategy of the proof is robust, and we present several potential extensions of the result.

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arXiv2025

Superlinear gradient growth for 2D Euler equation without boundary

We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus $\mathbb{T}^{2}$ and the whole plane $\mathbb{R}^{2}$. In the torus, whenever we have a steady state $\omega^*$ that is orbitally stable up to a translation and has a saddle point, we construct ${\tilde{\omega}}_0 \in C^\infty(\mathbb{T}^2)$ that is arbitrarily close to $\omega^*$ in $L^2$, such that superlinear growth of the vorticity gradient occurs for an open set of smooth initial data around ${\tilde{\omega}}_0$. This seems to be the first superlinear growth result which holds for an open set of smooth initial data (and does not require any symmetry assumptions on the initial vorticity). Furthermore, we obtain the first superlinear growth result for smooth and compactly supported vorticity in the plane, using perturbations of the Lamb-Chaplygin dipole.

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arXiv2025

Existence and stability of Sadovskii vortices: from patch to smooth vortices

We establish a scaling-invariant variational framework for steadily translating dipoles of the two-dimensional incompressible Euler equations. Specifically, we consider the maximization of the kinetic energy subject to constraints on the impulse and the Lp-norm (1 4/3. By removing the mass constraint, we obtain a unified scaling-invariant variational principle valid for all 1<p\leq\infty. As a consequence of the variational structure, we establish a Lyapunov-type stability result, demonstrating that the axis-touching geometry persists under small perturbations. Finally, we derive a quantitative bound on the horizontal center of mass of perturbed solutions, showing that they propagate at nearly the same speed as the underlying Sadovskii vortex.

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arXiv2025

MHS equilibria in the non-resistive limit to the randomly forced resistive magnetic relaxation equations

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity $\kappa>0$ and a force proportional to $\sqrt{\kappa}\ $ on the flat $d$-torus $\mathbb{T}^{d}$ for $d\geq 2$. We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium $B(x)$ in $H^{1}(\mathbb{T}^{d})$ with law $D(B)=\mu$ as a non-resistive limit $\kappa\to 0$ of statistically stationary solutions $B_{\kappa}(x,t)$. For $d=2$, the measure $\mu$ does not concentrate on any compact sets in $H^{1}(\mathbb{T}^{2})$ with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium $B(x)$ are almost surely not finite Fourier mode solutions.

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arXiv2025

Wellposedness of inviscid SQG in the half-plane

We consider the SQG equation without dissipation on the half-plane with Dirichlet boundary condition, and prove local wellposedness in the spaces $W^{3,p}$ and $C^{2,\beta}$ for any $1<p<\infty$ and $0<\beta<1$. We complement this wellposedness by showing that for generic $C^{\infty}_{0}$ initial data, the unique corresponding solution does not belong to $W^{3,\infty}$.

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arXiv2024

On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

For axisymmetric flows without swirl and compactly supported initial vorticity, we prove the upper bound of $t^{4/3}$ for the growth of the vorticity maximum, which was conjectured by Childress [Phys. D, 2008] and supported by numerical computations from Childress--Gilbert--Valiant [J. Fluid Mech. 2016]. The key is to estimate the velocity maximum by the kinetic energy together with conserved quantities involving the vorticity.

math.AP↗
arXiv2024

Stability of vortex quadrupoles with odd-odd symmetry

For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in $L^1$, we prove that it remains so for all times. The main difficulty is that the kinetic energy maximization problem in a quadrant -- the typical approach for establishing vortex stability -- lacks a solution, as the kinetic energy continues to increase when the vorticity escapes to infinity. We overcome this by taking dynamical information into account: finite-time desingularization result is combined with monotonicity of the first moment and a careful analysis of the interaction energies between vortices. The latter is achieved by new pointwise estimates on the Biot--Savart kernel and quantitative stability results for general interaction kernels. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other.

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arXiv2024

On existence of Sadovskii vortex patch: A touching pair of symmetric counter-rotating uniform vortex

The Sadovskii vortex patch is a traveling wave for the two-dimensional incompressible Euler equations consisting of an odd symmetric pair of vortex patches touching the symmetry axis. Its existence was first suggested by numerical computations of Sadovskii in [J. Appl. Math. Mech., 1971], and has gained significant interest due to its relevance in inviscid limit of planar flows via Prandtl--Batchelor theory and as the asymptotic state for vortex ring dynamics. In this work, we prove the existence of a Sadovskii vortex patch, by solving the energy maximization problem under the exact impulse condition and an upper bound on the circulation.

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arXiv2024

On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

It has been shown in our previous work that the incompressible and irresistive Hall- and electron-magnetohydrodynamic (MHD) equations are illposed on flat domains $M = \mathbb{R}^k \times \mathbb{T}^{3-k}$ for $0 \le k \le 2$. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in $\mathbb{R}^3$. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovski\v{i} operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.

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