We consider density solutions for gradient flow equations of the form u t = ∇ · ( γ ( u )∇ N( u )), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ ( u ) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ ( u ) = u . For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/ d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ ( u ) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

中文翻译：

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具有牛顿斥力和超线性迁移率的非局部相互作用模型的涡流形成

我们考虑 ut = ∇ · ( γ ( u )∇ N( u )) 形式的梯度流动方程的密度解，其中 N 是整个空间 ℝ d 中的牛顿排斥势，具有非线性凸迁移率 γ ( u ) = u α 和 α > 1。我们表明，对应于紧支持的初始数据的解决方案始终保持紧支持，导致移动自由边界，如线性迁移率情况 γ ( u ) = u 。对于线性流动性，有一个特殊的解决方案，即空间 u = c 1 t -1 中强度恒定的圆盘涡旋形式存在于一个球中，该球在时间上像 c 2 t 1/d 一样扩展，因此显示了不连续的前沿或冲击。我们目前的结果与 γ ( u ) = u α 形式的凹迁移率的情况形成鲜明对比，在 [10] 中研究了 0 < α < 1。那里，我们开发了一种粘度解的适定性理论，该理论在任何地方都是正的，而且在无穷远处显示出一条肥尾。在这里，我们还开发了粘度解的适定性理论，该理论在径向情况下会导致非常详细的分析，从而使我们能够展示等待时间现象。这是非线性退化扩散方程（例如多孔介质方程）的典型行为。我们还将构建显式自相似解，其表现出类似涡旋行为，在某些假设下表征一般径向解的长时间渐近。提出了基于黏度解理论的收敛数值方案，分析了它们的收敛速度。我们用数值模拟来补充我们的分析结果，说明经过验证的结果并展示一些未解决的问题。