In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced in \cite{ST-topo}. The main result demonstrates local existence and uniqueness of classical solutions in class $(\rho,u) \in H^{m+\alpha} \times H^{m+1}$ on the periodic domain $\mathbb{T}^n$, where $0<\alpha<2$ is the order of singularity of the topological communication kernel $\phi(x,y)$, and $m = m(n,\alpha)$ is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator \[ \mathcal{L}_\phi f(x) = \int_{\mathbb{T}^n} \phi(x,y) (f(y) - f(x) ) dy, \] which render proper a priori estimates and help stabilize viscous approximation of the system. In dimension 1, this result, in conjunction with the technique developed in \cite{ST-topo} gives global well-posendess in the natural space of data mentioned above.

中文翻译：

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集体行为的拓扑欧拉对齐模型的局部适定性

在本文中，我们解决了\cite{ST-topo} 中引入的多维拓扑欧拉对齐模型的适定性问题。主要结果证明了类 $(\rho,u) \in H^{m+\alpha} \times H^{m+1}$ 在周期域 $\mathbb{T}^ 中经典解的局部存在性和唯一性n$，其中$0<\alpha<2$为拓扑通信核$\phi(x,y)$的奇点阶数，$m = m(n,\alpha)$较大。我们的方法基于对拓扑对齐算子 \[ \mathcal{L}_\phi f(x) = \int_{\mathbb{T}^n} \phi(x,y) (f( y) - f(x) ) dy, \] 呈现正确的先验估计并帮助稳定系统的粘性近似。在维度 1 中，这个结果，