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Min Jun Jo

Publications and source records attributed to Min Jun Jo.

7 recordsLinked to original sources
arXiv2025

Sharper bounds on the box-counting dimension of singularities in the hyperdissipative Navier-Stokes system

We study upper bounds on the box-counting dimension of the set of potential singular points in suitable weak solutions to the 3D incompressible hyperdissipative Navier-Stokes system \begin{equation*} \partial_t u + (-Δ)^αu+(u\cdot \nabla)u+\nabla p = 0, \qquad \operatorname{div} u = 0, \end{equation*} for $α\in(1,5/4)$. Our main observation is that a classical iteration scheme developed in [11] and used in [27] to improve upper bounds for the full Laplacian case can be extended to the hyperdissipative case with properly chosen local quantities that are scale-invariant, despite non-locality of fractional Laplacian. This is achieved by matching up the correct orders of the temporal-spatial scales of the required estimates that effectively quantify $(-Δ)^α$ during the iterations. In particular, we adopt the hyperdissipative framework built in the recent breakthrough [5] where the upper bounds on the box-counting dimension of the set of potential singularities in $α$ are given by \begin{equation*} L(α)= \frac{15-2α-8α^2}{3} \quad \mbox{for}\quad 1<α<\frac{5}{4}. \end{equation*} In this paper, we generalize the iteration scheme [27] designed for $α=1$ to the case $1<α<5/4$, which leads to the newly established bound \begin{equation*} J(α)= \frac{36(3-α)(3+2α)(5-4α)}{-64α^3+272α^2-300α+369} \quad \mbox{for} \quad 1<α<\frac{5}{4}, \end{equation*} improving the aforementioned bound $L(α)$ obtained in [5].

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arXiv2025

Cusp Formation in Vortex Patches

We prove instantaneous cusp formation for any initial vortex patch with acute corners. This was conjectured to occur in the numerical literature.

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arXiv2024

Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the three fundamental domains

We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on $\bbR^2$ with respect to the buoyancy frequency $N$. Our target density of stratification is the sum of the large background linear profile with its slope $N$ and the small perturbation that could be both non-linear and non-monotone. Quantification in $N$ will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability. For the purely linear density stratification, our method robustly applies to the three fundamental domains $\bbR^2,$ $\bbT^2,$ and $\bbT\times[-1,1]$, improving both the previous result by Elgindi (On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation, Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on $\bbR^2$ and $\bbT^2$, and the study by Castro-Córdoba-Lear (Global existence of quasi-stratified solutions for the confined IPM equation. Archive for Rational Mechanics and Analysis, 232(1), 437-471, 2019) on $\bbT\times[-1,1]$. The obtained temporal decay rates to the stratified density on $\bbR^2$ and to the newly found asymptotic density profiles on $\bbT^2$ and $\bbT\times[-1,1]$ are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in $H^m$ for any integer $m\geq 4$, which we even relax to any positive number $m>3$ via a suitable anisotropic commutator estimate.

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arXiv2024

Non-convergence of the rotating stratified flows toward the quasi-geostrophic dynamics

The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.

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arXiv2022

Global well-posedness of the partially damped 2D MHD equations via a direct normal mode method for the anisotropic linear operator

We prove the global well-posedness of the 2D incompressible non-resistive MHD equations with a velocity damping term near the non-zero constant background magnetic field. To this end, we newly design a normal mode method of effectively leveraging the anisotropy of the linear propagator that encodes both the partially dissipative nature of the non-resistive MHD system and the stabilizing mechanism of the underlying magnetic field. Isolating new key quantities and estimating them with themselves in an entangling way via the eigenvalue analysis based on Duhamel's formulation, we establish the global well-posedness for any initial data $(v_0,B_0)$ that is sufficiently small in a space rougher than $H^{4}\cap L^1$. This improves the recent work in SIAM J. Math. Anal. 47, 2630-2656 (2015) where the similar result was obtained provided that $(v_0,B_0)$ was small enough in a space strictly embedded in $H^{20}\cap W^{6,1}$.

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