Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

TL;DR

This paper studies the evolution of Lagrangian graphs under mean curvature flow in pseudo-Euclidean space, establishing existence, bounds, decay estimates, and convergence to self-expanding solutions for related nonlinear equations.

Contribution

It introduces new results on the smooth solutions, derivative bounds, and convergence behavior of the mean curvature flow in pseudo-Euclidean space for a range of related equations.

Findings

Existence of smooth solutions for the parabolic equations.

Bounded derivatives for specific parameter values.

Convergence to self-expanding solutions.

Abstract

In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation \eqref{11} has a smooth solution $u(x,t)$ for three corresponding nonlinear equations between the Monge-Amp$\grave{e}$re type equation($\tau=0$) and the special Lagrangian parabolic equation($\tau=\frac{\pi}{2}$). Furthermore, we get the bound of $D^lu$, $l=\{3,4,5,\cdots\}$ for $\tau=\frac{\pi}{4}$ and the decay estimates of the higher order derivatives when $0<\tau<\frac{\pi}{4}$ and $\frac{\pi}{4}<\tau<\frac{\pi}{2}$. We also prove that $u(x,t)$ converges to smooth self-expanding solutions of \eqref{12}.