The nonlocal-interaction equation near attracting manifolds

<p style='text-indent:20px;'>We study the approximation of the nonlocal-interaction equation restricted to a compact manifold <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> embedded in <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula>, and more generally compact sets with positive reach (i.e. prox-regular sets). We show that the equation on <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula> can be approximated by the classical nonlocal-interaction equation on <inline-formula><tex-math id="M4">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> by adding an external potential which strongly attracts to <inline-formula><tex-math id="M5">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. The proof relies on the Sandier–Serfaty approach [<xref ref-type="bibr" rid="b23">23</xref>,<xref ref-type="bibr" rid="b24">24</xref>] to the <inline-formula><tex-math id="M6">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-convergence of gradient flows. As a by-product, we recover well-posedness for the nonlocal-interaction equation on <inline-formula><tex-math id="M7">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, which was shown [<xref ref-type="bibr" rid="b10">10</xref>]. We also provide an another approximation to the interaction equation on <inline-formula><tex-math id="M8">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>, based on iterating approximately solving an interaction equation on <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb{R}}^d $\end{document}</tex-math></inline-formula> and projecting to <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal{M}} $\end{document}</tex-math></inline-formula>. We show convergence of this scheme, together with an estimate on the rate of convergence. Finally, we conduct numerical experiments, for both the attractive-potential-based and the projection-based approaches, that highlight the effects of the geometry on the dynamics.</p>

[1]  J. Carrillo,et al.  The role of a strong confining potential in a nonlinear Fokker–Planck equation , 2018, Nonlinear Analysis.

[2]  J. Rataj,et al.  On the structure of sets with positive reach , 2016, 1604.08841.

[3]  Seung-Yeal Ha,et al.  A Second-Order Particle Swarm Model on a Sphere and Emergent Dynamics , 2019, SIAM J. Appl. Dyn. Syst..

[4]  S. Lisini Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces , 2009 .

[5]  P. Sternberg,et al.  Convergence of a Particle Method for Diffusive Gradient Flows in One Dimension , 2016, SIAM J. Math. Anal..

[6]  Matthias Hein,et al.  Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator , 2018, Found. Comput. Math..

[7]  Seung‐Yeal Ha,et al.  Particle and Kinetic Models for Swarming Particles on a Sphere and Stability Properties , 2018, Journal of Statistical Physics.

[8]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[9]  Katy Craig,et al.  Convergence of Regularized Nonlocal Interaction Energies , 2015, SIAM J. Math. Anal..

[10]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[11]  S. Serfaty,et al.  Gamma‐convergence of gradient flows with applications to Ginzburg‐Landau , 2004 .

[12]  Seung‐Yeal Ha,et al.  Emergent behaviors of Cucker–Smale flocks on the hyperboloid , 2020, 2007.02556.

[13]  Sara Daneri,et al.  Lecture notes on gradient flows and optimal transport , 2010, Optimal Transport.

[14]  Jos'e A. Carrillo,et al.  A well-posedness theory in measures for some kinetic models of collective motion , 2009, 0907.3901.

[15]  J. Carrillo,et al.  Nonlocal-interaction equations on uniformly prox-regular sets , 2014, 1405.1111.

[16]  Razvan C. Fetecau,et al.  An Intrinsic Aggregation Model on the Special Orthogonal Group SO(3): Well-posedness and Collective Behaviours , 2020, Journal of Nonlinear Science.

[17]  J. A. Carrillo,et al.  The derivation of swarming models: Mean-field limit and Wasserstein distances , 2013, 1304.5776.

[18]  L. Ambrosio,et al.  A User’s Guide to Optimal Transport , 2013 .

[19]  Sylvia Serfaty,et al.  Gamma-convergence of gradient flows on Hilbert and metric spaces and applications , 2011 .

[20]  T. Laurent,et al.  Lp theory for the multidimensional aggregation equation , 2011 .

[21]  E. Ostermann Convergence of probability measures , 2022 .

[22]  R. Fetecau,et al.  Self-organization on Riemannian manifolds , 2018, Journal of Geometric Mechanics.

[23]  D. Slepčev,et al.  Nonlocal Interaction Equations in Environments with Heterogeneities and Boundaries , 2015 .