Given an entire \(C^2\) function u on \(\mathbb {R}^n\) , we consider the graph of Du as a Lagrangian submanifold of \(\mathbb {R}^{2n}\) , and deform it by the mean curvature flow in \(\mathbb {R}^{2n}\) . This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of \((I+(D^2 u)^2)^{-1}D^2 u\) . Such results were previously known only under the stronger assumption of positivity of \(D^2 u\) .
