Existence and weak evolution theory for the coupled fractional mean-curvature flow

Establish general existence, including short-time existence, for the coupled system of geometric laws governing multiple nested fronts in the strongly nonlocal fractional Allen–Cahn limit, and develop a weak formulation that remains meaningful after singularity formation.

Background

The paper identifies a coupled geometric evolution law for finitely many strictly nested fronts, in which each front moves according to its fractional mean curvature together with nonlocal interaction terms generated by the other fronts. The main convergence theorem is conditional: it assumes that this coupled system already admits a smooth evolution whose fronts remain strictly nested on the time interval under consideration.

The authors explicitly state that they do not establish general existence for the multiple-front coupled system. They further identify short-time existence and a weak formulation capable of handling singularity formation as unresolved objectives. Such a weak theory is needed because nonlocal curvature flows can develop singularities, beyond the regime where the smooth evolution assumed by the convergence theorem is available.

References

We do not establish general existence for the coupled system of multiple fronts. Nevertheless, Theorem~\ref{thm:main_result} identifies the coupled motion of every front whenever a smooth, strictly nested evolution exists. Establishing short-time existence for eq:velocity-intro and developing a weak formulation that accounts for singularity formation are objectives of a future paper.

eq:velocity-intro:

vi(t,x)=c0(12H2s(Ωti)(x)−∑j=1i−1∫(Ωtj‾)cdz∣z−x∣n+2s+∑j=i+1N∫Ωtjdz∣z−x∣n+2s),x∈Γti,v_i(t,x)=c_0\left(\frac{1}{2} H_{2s}(\Omega^i_t)(x)-\sum_{j=1}^{i-1} \int_{(\overline{\Omega_t^j})^c} \frac{dz}{|z-x|^{n+2s}} +\sum_{j=i+1}^{N} \int_{\Omega^j_t} \frac{dz}{|z-x|^{n+2s}}\right),\qquad x\in \Gamma_t^i,

— Interacting fronts in the strongly nonlocal Allen--Cahn equation  (2609.35666 - Hasani et al., 28 Sep 2026) in Introduction, paragraph following the statement of Theorem 1.1