Dongkwang Kim | Searcharxiv

arXiv2026

A gradient flow for the porous medium equations with Dirichlet boundary conditions

We consider the gradient flow structure of the porous medium equations with non-negative constant Dirichlet boundary conditions. We construct weak solutions to the equations via the minimizing movement scheme by considering an entropy functional with respect to $Wb_2$ distance, which is a modified Wasserstein distance introduced by Figalli and Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130]. Furthermore, the constructed solutions are characterized as curves of maximal slope in a suitable sense.

math.AP↗

arXiv2024

Global boundedness and blow-up in a repulsive chemotaxis-consumption system in higher dimensions

This paper investigates the repulsive chemotaxis-consumption model \begin{align*} \partial_t u &= \nabla \cdot (D(u) \nabla u) + \nabla \cdot (u \nabla v), \\ 0 &= Δv - uv \end{align*} in an $n$-dimensional ball, $n \ge 3$, where the diffusion coefficient $D$ is an appropriate extension of the function $0\leξ\mapsto(1+ξ)^{m-1}$ for some $m>0$. Under the boundary conditions \begin{equation*} ν\cdot (D(u) \nabla u + u \nabla v) = 0 \quad\text{ and }\quad v = M>0,\end{equation*} we first demonstrate that for $m > 1$, or $m = 1$ with $0 < M < 2/(n-2)$, the system admits globally defined classical solutions that are uniformly bounded in time for any choice of sufficiently smooth radial initial data. This result is further extended to the case $0<m<1$ when $M$ is chosen to be sufficiently small, depending on the initial conditions. In contrast, it is shown that for $0 < m < \frac{2}{n}$, the system exhibits blow-up behavior for sufficiently large $M$.

math.AP↗