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The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

Huan Wu, Shiqing Zhang

TL;DR

This work establishes the existence and stability of viscosity solutions for perturbed contact Hamilton–Jacobi equations on a compact manifold. By embedding the problem in the weak KAM framework and employing backward solution semigroups, the authors show that a viscosity solution $u_-$ of the unperturbed equation persists under small perturbations as a unique, uniformly convergent solution $u_-^{\varepsilon}$, provided Lyapunov stability and a nondegeneracy condition hold. The analysis leverages the closeness of perturbed and unperturbed Lagrangians, quantitative semigroup estimates, and the implicit action function to control deviations and guarantee convergence. Under a positivity condition on $\partial H/\partial u$ over the calibrated set $\Lambda_{u_-}$, the perturbed problem yields a unique $u_-^{\varepsilon}$ near $u_-$ that retains local Lyapunov asymptotic stability, highlighting robustness of stationary states in dissipative Hamiltonian systems. These results contribute to the understanding of perturbations in contact Hamilton–Jacobi dynamics and extend weak KAM methods to $u$-dependent Hamiltonians with stability guarantees.

Abstract

We consider a contact Hamiltonian $H(x,p,u)$ with certain dependence on the contact variable $u$. If $u_{-}$ is a viscosity solution of the contact Hamilton-Jacobi equation \[H(x,D_{x}u(x),u(x))=0,\quad x\in M,\] and $u_{-}$ is locally Lyapunov asymptotically stable, we will prove that the perturbed equation \[H(x,D_{x}u(x),u(x))+\varepsilon P(x,D_{x}u(x),u(x))=0,\quad x\in M,\] does exist viscosity solution $u_{-}^{\varepsilon}$ which converges uniformly to $u_{-}$, as perturbation parameter $\varepsilon$ converges to 0. Moreover, we give a case that in a neighborhood of viscosity solution $u_-$, the perturbed equation has an unique viscosity solution $u_{-}^{\varepsilon}$. Furthermore, $u_{-}^{\varepsilon}$ keeps locally Lyapunov asymptotically stability.

The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

TL;DR

This work establishes the existence and stability of viscosity solutions for perturbed contact Hamilton–Jacobi equations on a compact manifold. By embedding the problem in the weak KAM framework and employing backward solution semigroups, the authors show that a viscosity solution of the unperturbed equation persists under small perturbations as a unique, uniformly convergent solution , provided Lyapunov stability and a nondegeneracy condition hold. The analysis leverages the closeness of perturbed and unperturbed Lagrangians, quantitative semigroup estimates, and the implicit action function to control deviations and guarantee convergence. Under a positivity condition on over the calibrated set , the perturbed problem yields a unique near that retains local Lyapunov asymptotic stability, highlighting robustness of stationary states in dissipative Hamiltonian systems. These results contribute to the understanding of perturbations in contact Hamilton–Jacobi dynamics and extend weak KAM methods to -dependent Hamiltonians with stability guarantees.

Abstract

We consider a contact Hamiltonian with certain dependence on the contact variable . If is a viscosity solution of the contact Hamilton-Jacobi equation and is locally Lyapunov asymptotically stable, we will prove that the perturbed equation does exist viscosity solution which converges uniformly to , as perturbation parameter converges to 0. Moreover, we give a case that in a neighborhood of viscosity solution , the perturbed equation has an unique viscosity solution . Furthermore, keeps locally Lyapunov asymptotically stability.
Paper Structure (9 sections, 14 theorems, 96 equations)

This paper contains 9 sections, 14 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. A brief introduction to contact Hamilton-Jacobi equation
  3. main results
  4. Notations
  5. Preliminaries
  6. The Weak KAM Theorem
  7. Some fundamental results for equation \ref{['eqth1']}
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.1

Suppose $P\in C_{0}^{3}(T^{*}M\times\mathbf{R},\mathbf{R})$ and for all $(x,p,u)\in T^*M\times\mathbf{R},\ |P(x,p,u)|\leqslant1$, $u_{-}\in \mathcal{S}^{-}$. If $u_-$ is locally Lyapunov asymptotically stable, then given $\delta>0$, there has $\varepsilon_{\delta}>0$ such that, for all $\varepsilon\

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • ...and 14 more