arXiv subjects

Woojae Lee

Publications and source records attributed to Woojae Lee.

3 recordsLinked to original sources

arXiv2026

On a generalized incompressible model in two dimensions

We analyze a generalized incompressible model proposed by Ohkitani [19]. This model is based on the observation that the two-dimensional Burgers' equation can be related to the incompressible Navier-Stokes equations by rotating the velocity gradient by 90 degrees. We present several results with initial data in $H^{3}$. First of all, we examine the inviscid model and show the existence and uniqueness of a local-in-time solution that blows up in finite time if the initial vorticity contains a negative part. In the presence of viscosity, we show the existence of a unique global-in-time solution without requiring a sign condition on the initial vorticity, establish the long-time behavior of the difference between two solutions, and derive temporal decay rates for the velocity field when the initial vorticity is non-positive.

math.AP↗

arXiv2025

Shock-type singularity of the hyperbolic-parabolic chemotaxis system

This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated with the shock-type structure, which was numerically observed by Filbet, Lauren{\c{c}}ot, and Perthame \cite{filbet2005derivation} and Filbet and Shu \cite{filbet2005approximation}. We construct the blow-up profile in a 1D HPC system on $\mathbb{R}$ as follows: The blow-up profile is stable in the sense of $H^m$ topology ($m\geq 5$) prior to the occurrence of the singularity. For the first singularity, while the density and velocity $(\rho, u)$ of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration $\phi$ has $C^2$ regularity. However, the density and velocity with $C^ {\frac{1}{3}}$ regularity exhibit a cusp singularity at a unique blow-up point, the location and time of which are explicitly estimated. Furthermore, the HPC system is $C^1$ differentiable except in any neighborhood of the blow-up point.

math.AP↗

arXiv2024

Global existence and asymptotic stability for the Toner-Tu model of flocking

This paper deals with the Toner-Tu (TT) model, which is a hydrodynamic model describing the collective motion of numerous self-propelled agents. We analytically study the global-in-time well-posedness of the TT model near the steady-state solution in the ordered phase. We also show the large-time behavior of solutions showing that the steady-state solution is polynomially stable in a Sobolev space in the sense that solutions that are initially close to that steady state converge to that at least polynomially fast as time tends to infinity. Moreover, we investigate the variant of the TT model which describes the dynamics of the actin filament.

math.AP↗