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- W2255657988 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 $0 leq t < 1$." @default.
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- W2255657988 date "2015-07-28" @default.
- W2255657988 modified "2023-09-27" @default.
- W2255657988 title "The regularity of the boundary of a multidimensional aggregation patch" @default.
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