Delta-ring

Abstract: We study similarity solutions to the multidimensional aggregation equation $u_t+{div}(uv)=0$, $v=- abla K*u$ with general power-law kernels K such that $ abla K(x)=x|x|^{\alpha-2}$, $\alpha\in(2-d,2)$ We analyze the equation in different regimes of the parameter $\alpha$ In the case when $\alpha\in[4-d,2)$, we give a characterization of all the “first-kind” radially symmetric similarity solutions We prove that any such solution is a linear combination of a delta ring and a delta mass at the origin On the other hand, when $\alpha\in(2-d,4-d)$, we show that there exist multi delta-ring similarity solutions in $\mathbb{R}^d$ In particular, our results imply that multi delta-ring similarity solutions exist in three dimensions if $\alpha$ is just a little bit below 1

Abstract: We study similarity solutions to the multidimensional aggregation equation $u_t+\Div(uv)=0$, $v=- abla K*u$ with general power-law kernels $K(x)=|x|^\alpha,\alpha\in (2-d,2)$. We analyze the equation in different regimes of the parameter $\alpha$. In the case when $\alpha\in [4-d,2)$, we give a characterization all the "first kind" radially symmetric similarity solutions. We prove that any such solution is a linear combination of a delta ring and a delta mass at the origin. On the other hand, when $\alpha\in (2-d,4-d)$, we show that there exist multi delta-ring similarity solutions in $R^d$. In particular, our results imply that multi delta-ring similarity solutions exist in 3D if $\alpha$ is just a little bit below 1.