  In this paper, we introduce a geometric flow for Lagrangian submanifolds in a
K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a
gradient flow for volume. The stationary solutions are the Hamiltonian
stationary Lagrangian submanifolds. The flow is not strictly parabolic but it
corresponds to a fourth order strictly parabolic scalar equation in the
cotangent bundle of the submanifold via Weinstein's Lagrangian neighborhood
theorem. For any compact initial Lagrangian immersion, we establish short-time
existence, uniqueness, and higher order estimates when the second fundamental
forms are uniformly bounded up to time $T$.
