Lyapunov stability and uniqueness problems for Hamilton-Jacobi equations without monotonicity

TL;DR

This work advances the theory of the contact Hamilton-Jacobi equation $w_t+w_t+H(x,Dw,w)=0$ on a compact manifold by developing a non-Lyapunov, variational framework based on Mather measures and weak KAM theory to address Lyapunov stability and uniqueness of stationary viscosity solutions in the non-monotone setting. By constructing and exploiting action-minimizing structures, Mañé sets, and the associated Mather measures, the authors derive explicit criteria: the sign of $\int \frac{\partial H}{\partial u}\, d\mu$ over Mather measures governs stability vs. instability, and a positive average over all Mather measures yields at most one viscosity solution, with global asymptotic stability when it holds for every measure. The paper also develops the necessary Mather/weak KAM theory for non-monotone contact Hamiltonians, including a decomposition theorem for the Mañé set and a detailed treatment of calibrated curves and $1$-graphs, providing both conceptual and technical tools for future analysis. Through illustrative examples on the unit circle, the authors demonstrate how parameter choices govern stability regimes and uniqueness outcomes, highlighting the practical impact for nonlinear, non-monotone Hamilton-Jacobi dynamics. Collectively, these results extend action-minimizing methods to a broader class of Hamiltonians, offering robust criteria for stability and uniqueness without relying on auxiliary Lyapunov functions.

Abstract

We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where $M$ is a compact manifold, $H:T^*M\times R\to R$, $H=H(x,p,u)$ satisfies Tonelli conditions in $p$ and the Lipschitz condition in $u$. This work mainly concerns with the Lyapunov stability (including asymptotic stability, and instability) and uniqueness of stationary viscosity solutions of the equation. A criterion for stability and a criterion for instability are given. We do not utilize auxiliary functions and thus our method is different from the classical Lyapunov's direct method. We also prove several uniqueness results for stationary viscosity solutions. The Hamiltonian $H$ has no concrete form and it may be non-monotonic in the argument $u$, where the situation is more complicated than the monotonic case. Several simple but nontrivial examples are provided, including the following equation on the unit circle \[ w_t(x,t)+\frac{1}{2}w^2_x(x,t)-a\cdot w_x(x,t)+(\sin x+b)\cdot w(x,t)=0,\quad x\in \mathbf{S}, \] where $a$, $b\in R$ are parameters. We analyze the stability, and instability of the stationary solution $w=0$ when parameters vary, and show that $w=0$ is the unique stationary solution when $a=0$, $b>1$ and $a\neq0$, $b\geqslant 1$. The sign of the integral of $\frac{\partial H}{\partial u}$ with respect to the Mather measure of the contact Hamiltonian system generated by $H$ plays an essential role in the proofs of aforementioned results. For this reason, we first develop the Mather and weak KAM theories for contact Hamiltonian systems in this non-monotonic setting. A decomposition theorem of the Mañé set is the main result of this part.

Theorems & Definitions (86)

• Definition 1.1
• Remark 1.1
• Remark 1.2
• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Example 1.1
• Theorem 1.4
• Remark 1.3