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Local well-posedness of the topological Euler alignment models of collective behavior. (English) Zbl 1451.92350

Summary: In this paper we address the problem of well-posedness of multi-dimensional topological Euler-alignment models introduced by the second author and E. Tadmor [SIAM J. Math. Anal. 52, No. 6, 5792–5839 (2020; Zbl 1453.92371)]. 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 \(\varphi (x, y)\), and \(m=m(n,\alpha)\) is large. Our approach is based on new sharp coercivity estimates for the topological alignment operator \(\mathcal{L}_{\phi}f(x)=\int_{\mathbb{T}^n}\phi(x,y)(f(y)-f(x))\text{d}y\), which render proper a priori estimates and help stabilize viscous approximation of the system.

MSC:

92D50 Animal behavior
54H99 Connections of general topology with other structures, applications

Citations:

Zbl 1453.92371
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References:

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