Abstract
We study the well-posedness of a class of nonlocal-interaction equations on general domains $\Omega\subset \mathbb{R}^{d}$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $\lambda$-geodesically convex interaction and external potentials, the nonlocal-interaction equationshave unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy.We use these results to investigate on which domains and for which potentials the solutions aggregate to a single point as time goes to infinity.Our approach is based on the theory of gradient flows in spaces of probability measures.
Highlights
We study a continuum model of agents interacting via a potential W and subject to an external potential V confined to a closed subset Ω ⊂ Rd
Our main result is the well-posedness of weak measure solutions, described in Definition 1.1, of the nonlocal-interaction equation (1.1) on uniformly prox-regular domains
After establishing the well-posedness of gradient-flow solutions we show the well-posedness of weak measure solutions
Summary
Our main result is the well-posedness of weak measure solutions, described in Definition 1.1, of the nonlocal-interaction equation (1.1) on uniformly prox-regular domains. The main results of this paper is the well-posedness of weak measure solutions: existence and stability, with arbitrary initial data We establish it using an approximation scheme and the theory of gradient flows in spaces of probability measures. Note that the conditions (LA1) and (LA3) are satisfied whenever V and W are C2 functions on Rd, which is the case in many practical applications We show in this setting the following theorem about existence and stability for weak measure solutions for initial data with compact support.
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