Biological cells collapsing toward a single point behave like a smooth, flowing liquid in three dimensions, but they snap into existence instantly in two.
The Keller-Segel system models how things like bacteria or chemicals clump together until they reach a point of infinite density called a singularity. In a flat, two-dimensional world, these collapses happen in an instant jump. This new research shows that in our three-dimensional world, the collapse happens continuously over time instead. The way matter aggregates changes fundamentally based on the number of dimensions it occupies. This discovery provides a more accurate way to model how life and matter concentrate in the real world.
Formation and Behavior of Dirac Singularities in the Parabolic-Elliptic Keller-Segel System in Dimensions $n\geq 3$
arXiv · 2605.00110
We consider nonnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel system \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\\ &0=\Delta v -\mu + u , \\ \end{array}\right. \end{align*} where $\mu$ is the spatial average of $u$, under homogeneous Neumann boundary conditions in a ball in $\mathbb R^n$ for $n\geq 3$. In two dimensions, it is well established that solutions blowing up in finite time converge to a Dirac p