Large-time Behavior for a Cutoff Level-Set Mean Curvature G-equation with Source term

TL;DR

This work analyzes the large-time behavior and regularity of a cutoff level-set mean curvature G-equation with non-negative source, in both periodic and radially symmetric contexts. Using viscosity solutions and a structure-based monotonicity framework centered on the Aubry set, it proves uniform convergence to ergodic stationary profiles and derives a radial representation formula for the limiting state. Regularity results include local Lipschitz continuity in the periodic setting (with wind vanishing) and global Lipschitz in time plus radial-space Lipschitz in the radially symmetric case, aided by a convex Hamiltonian in the radial reduction. The findings illuminate front propagation under cutoff nonlinearity, with implications for homogenization, combustion modeling, and crystal growth dynamics.

Abstract

We consider a cutoff level-set mean curvature G-equation with a non-negative source term. In particular, we study the large-time behavior of this fully nonlinear degenerate parabolic partial differential equation in two settings: periodic and radially symmetric. Due to the non-coercivity and non-convexity of the Hamiltonian, we are not able to use the standard Hamilton-Jacobi theory and instead rely on the inherent structure to find a monotonicity property and corresponding uniqueness set. Lastly, we discuss the local regularity of the solutions, as well as a representation formula in the radial symmetric setting.

Figures (2)

• Figure 3.1: Trajectories from the right.
• Figure 3.2: Cone of admissible trajectories.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1
• Remark 2
• Lemma 2.1
• proof
• Proposition 2.2
• proof : Proof of Theorem \ref{['periodic largetime']}
• Remark 3