On short time existence of Lagrangian mean curvature flow

We consider a short time existence problem motivated by a conjecture of Joyce (Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. arXiv:1401.4949 , 2014). Specifically we prove that given any compact Lagrangian $$L\subset \mathbb {C}^n$$ with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains L as $$t \searrow 0$$ as varifolds, and smoothly locally away from the singularities.