[Paper Review] Smooth subsolutions of the discounted Hamilton-Jacobi equations

Huang, Xiyao, Liang Jin+2 more|arXiv (Cornell University)|Jul 21, 2020

Quantum chaos and dynamical systems19 references4 citations

TL;DR

This paper constructs $C^{1,1}$ subsolutions to the discounted Hamilton-Jacobi equation $\lambda u + H(x, du) = 0$ on a compact Riemannian manifold, showing they are solutions precisely on the projected Aubry set. Under additional hyperbolicity assumptions, the smoothness of these subsolutions improves, enabling identification of the global attractor of the conformally symplectic flow and control of the convergence speed of Lax-Oleinik semigroups.

ABSTRACT

For the discounted Hamilton-Jacobi equation,$$λu+H(x,d_x u)=0, \ x \in M, $$we construct $C^{1,1}$ subsolutions which are indeed solutions on the projected Aubry set. The smoothness of such subsolutions can be improved under additional hyperbolicity assumptions. As applications, we can use such subsolutions to identify the maximal global attractor of the associated conformally symplectic flow and to control the convergent speed of the Lax-Oleinik semigroups

Motivation & Objective

• To construct smooth subsolutions to the discounted Hamilton-Jacobi equation that are solutions on the projected Aubry set.
• To improve the regularity of these subsolutions under hyperbolicity assumptions.
• To apply the constructed subsolutions to identify the maximal global attractor of the conformally symplectic flow.
• To control the convergence speed of the Lax-Oleinik semigroup using the subsolutions.

Proposed method

• Uses the variational principle of $\lambda$-dominated functions to define the set $\mathcal{S}^{-}$ of subsolutions.
• Constructs $C^{1,1}$ subsolutions via a limiting process from $\lambda$-dominated functions.
• Proves that these subsolutions solve the discounted HJ equation on the projected Aubry set $\mathcal{A}$.
• Applies the theory of weak KAM solutions and global minimizers to relate the subsolutions to the viscosity solution $u^{-}$.
• Employs the Legendre transformation and Tonelli Lagrangian $L$ to link the Hamiltonian and variational formulations.
• Uses upper semicontinuity of the Aubry set under perturbations of $\lambda$ and $L$ to establish stability of the results.

Experimental results

Research questions

• RQ1 Can $C^{1,1}$ subsolutions be constructed for the discounted Hamilton-Jacobi equation that are solutions on the projected Aubry set?
• RQ2 How does additional hyperbolicity improve the regularity of such subsolutions?
• RQ3 Can these subsolutions be used to identify the maximal global attractor of the conformally symplectic flow?
• RQ4 What is the role of the viscosity solution $u^{-}$ in characterizing the Aubry set via global minimizers?
• RQ5 How do the Lax-Oleinik semigroups converge, and can this be controlled using the constructed subsolutions?

Key findings

• The viscosity solution $u^{-}$ of the discounted HJ equation is the pointwise supremum of all $C^{\infty}$ subsolutions in $\mathcal{S}^{-}$, establishing a duality between smooth subsolutions and the viscosity solution.
• The constructed $C^{1,1}$ subsolutions solve the discounted HJ equation exactly on the projected Aubry set $\mathcal{A}$, linking subsolution theory to the Aubry-Mather structure.
• Under hyperbolicity assumptions, the regularity of the subsolutions can be improved beyond $C^{1,1}$, suggesting stronger structural control.
• The Aubry set $\widetilde{\mathcal{A}}$ is characterized as the union of all globally calibrated curves, and $\mathcal{A} = \pi(\widetilde{\mathcal{A}})$ is a Lipschitz graph over $M$, confirming its intrinsic regularity.
• The Lax-Oleinik semigroup $\mathcal{T}^{-}_{t}$ converges to $u^{-}$ as $t \to \infty$, and the convergence rate is controlled by the decay of $F_{\gamma}(t) \to 0$, with $F_{\gamma}(t)$ bounded by $\epsilon$ for large $t$.
• The Aubry set is upper semicontinuous with respect to $\lambda$ and $L$ in the Hausdorff topology on $TM$, ensuring stability of the invariant set under perturbations.