  It is known that minimal Lagrangians in K\"ahler--Einstein manifolds of
non-positive scalar curvature are linearly stable under Hamiltonian
deformations. We prove that they are also stable under the Lagrangian mean
curvature flow, and therefore establish the equivalence between linear
stability and dynamical stability.
  Specifically, if one starts the mean curvature flow with a Lagrangian which
is $C^1$-close and Hamiltonian isotopic to a minimal Lagrangian, the flow
exists smoothly for all time, and converges to that minimal Lagrangian. Due to
the work of Neves [Ann. of Math. 2013], this cannot be true for
$C^0$-closeness.
