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Deokwoo Lim

Publications and source records attributed to Deokwoo Lim.

8 recordsLinked to original sources
arXiv2026

On the upper bound for the vorticity growth of bi-rotational Euler flows without swirl

For $d\geq 4$, we consider incompressible Euler flows in $\mathbb{R}^{d}$ with bi-rotational symmetry and without swirl. Our first result gives the local wellposedness of the Yudovich-type solution. The second result provides global wellposedness up to $d\leq 6$. In particular, it shows that the rate of growth of the vorticity maximum coincides with the rate from axisymmetric flows without swirl, which was obtained in the paper by the second author and Jeong (Arch. Ration. Mech. Anal. 249(3):32, 2025) and Shao--Wei--Zhang (Acta Math. Sin. (Engl. Ser.), 42(3):663-679, 2026).

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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.

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arXiv2024

On global regularity of some bi-rotational Euler flows in $\mathbb{R}^{4}$

In this paper, we consider incompressible Euler flows in $ \mathbb{R}^{4} $ under bi-rotational symmetry, namely solutions that are invariant under rotations in $\mathbb{R}^{4}$ fixing either the first two or last two axes. With the additional swirl-free assumption, our first main result gives local wellposedness of Yudovich-type solutions, extending the work of Danchin [Uspekhi Mat. Nauk 62(2007), no.3, 73-94] for axisymmetric flows in $\mathbb{R}^{3}$. The second main result establishes global wellposedness under additional decay conditions near the axes and at infinity. This in particular gives global regularity of $C^{\infty}$ smooth and decaying Euler flows in $\mathbb{R}^{4}$ subject to bi-rotational symmetry without swirl.

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arXiv2024

Global regularity of some axisymmetric, single-signed vorticity in any dimension

We consider incompressible Euler equations in any dimension $ d\geq3 $ imposing axisymmetric symmetry without swirl. While the global regularity of smooth flows in this setting has been well-known in $ d=3 $, the same question in higher dimensions $ d\geq4 $ remains unsolved. Recently, global regularity for the case $ d=4 $ with some extra decay assumption on vorticity is obtained by proving global estimate of the radial velocity. Now we prove that the vorticity with single-sign and a similar decay assumption is globally regular for any $ d\geq4 $. This is due to pointwise decay estimate of radial velocity in sufficiently large radial distance, which depends on time. The result is of confinement type for support growth, which is going back to Marchioro [Comm. Math. Phys., 164 (1994) 507-524] and Iftimie--Sideris--Gamblin [Comm. Partial Differential Equations, 24 (1999) 1709-1730] for $ \mathbb{R}^{2} $. In particular, we follow the approach of Maffei--Marchioro [Rend. Sem. Mat. Univ. Padova, 105 (2001) 125-137] for $ d=3 $ so that we generalize the confinement into any dimension.

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arXiv2022

Global regularity for some axisymmetric Euler flows in $\mathbb{R}^{d}$

We consider axisymmetric Euler flows without swirl in $\mathbb{R}^{d}$ with $d\geq 4$, for which the global regularity of smooth solutions is an open problem. When $d = 4$, we obtain global regularity under the assumption that the initial vorticity satisfies some decay at infinity and is vanishing at the axis. Assuming further that the initial vorticity is of one sign guarantees global regularity for $d\leq 7$.

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arXiv2022

Stability of monotone, non-negative, and compactly supported vorticities in the half cylinder and infinite perimeter growth for patches

We consider the incompressible Euler equations in the half cylinder $ \mathbb{R}_{>0}\times\mathbb{T}$. In this domain, any vorticity which is independent of $x_2$ defines a stationary solution. We prove that such a stationary solution is nonlinearly stable in a weighted $L^{1}$ norm involving the horizontal impulse, if the vorticity is non-negative and non-increasing in $x_1$. This includes stability of cylindrical patches $\{x_{1}<α\},\; α>0$. The stability result is based on the fact that such a profile is the unique minimizer of the horizontal impulse among all functions with the same distribution function. Based on stability, we prove existence of vortex patches in the half cylinder that exhibit infinite perimeter growth in infinite time.

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arXiv2021

Stability of radially symmetric, monotone vorticities of 2D Euler equations

We consider the incompressible Euler equations in $R^2$ when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity produces stability in some weighted norm related to the angular impulse. For instance, it covers the cases of circular vortex patches and Gaussian distributions. Our stability does not depend on $L^\infty$-bound or support size of perturbations. The proof is based on the fact that such a radial monotone distribution minimizes the impulse of functions having the same level set measure.

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