  We consider a short time existence problem motivated by a conjecture of
Joyce. 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.

Analysis of PDEs;Differential Geometry