Local well-posedness of the topological Euler alignment models of collective behavior

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.