The regularity of the boundary of a multidimensional aggregation patch

TL;DR

This paper proves that for a specific aggregation equation with initial data as a smooth bounded domain, the evolving domain remains smooth of class C^{1+γ} over time, maintaining boundary regularity up to a critical time.

Contribution

It establishes the preservation of boundary regularity for solutions of the aggregation equation with smooth initial domains, extending understanding of boundary behavior in such PDEs.

Findings

The solution exists and is explicitly given up to time t=1.

The boundary of the domain remains C^{1+γ} for all 0 ≤ t < 1.

Boundary regularity is preserved during evolution.

Abstract

Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$ \frac{\partial \rho}{\partial t} + \operatorname{div}(\rho v) =0, v = -\nabla N * \rho$$ with initial data $\rho(x,0) = \chi_{D_0}$, where $\chi_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < \gamma < 1$ and take $D_0$ to be a bounded $C^{1+\gamma}$ domain (a domain with smooth boundary of class $C^{1+\gamma}$). Then we have Theorem: If $D_0$ is a $C^{1 + \gamma}$ domain, then the initial value problem above has a solution given by $$\rho(x,t) = \frac{1}{1 -t} \chi_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + \gamma}$ domain for all $0 \leq t < 1$.