Commutation Properties of Semi-groups for Contact Type Hamilton-Jacobi Equation

1. Problem Statement

The paper addresses the commutation properties of Lax-Oleinik semi-groups associated with contact type Hamilton-Jacobi equations.

• Context: While the commutation of semi-groups for standard (normal) Hamilton-Jacobi equations has been established (notably by Barles and Tourin, linking it to the solvability of multi-time equations and weak KAM theory), the specific case for contact type equations (where the Hamiltonian depends explicitly on the solution $u$) remained less systematically investigated.
• The Equation: The study focuses on the Cauchy problem:
  $$\begin{cases} u_t(x, t) + H(x, Du(x, t), u(x, t)) = 0 & \text{in } \mathbb{R}^n \times (0, \infty) \\ u(x, 0) = u_0(x) \end{cases}$$
  where $H$ is a contact Hamiltonian.
• Core Question: Under what conditions do the semi-groups $S_H(t)$ and $S_F(t)$ generated by two different contact Hamiltonians $H$ and $F$ commute? That is, when does $S_H(\lambda) \circ S_F(\mu) = S_F(\mu) \circ S_H(\lambda)$ hold?

2. Methodology

The author employs a combination of viscosity solution theory, variational principles, and Jacobi bracket geometry.

• Viscosity Solutions & Variational Principles:

  • The paper establishes the existence and uniqueness of Lipschitz viscosity solutions using the implicit variational principle (Proposition 1.2).
  • It utilizes the Lax-Oleinik semi-group defined via an infimum over curves involving the Legendre transform $H^*$ of the Hamiltonian.
  • The comparison principle is used extensively to prove uniqueness and convergence of approximations.

• Jacobi Brackets vs. Poisson Brackets:

  • Unlike previous works on commuting Tonelli Hamiltonians (which rely on Poisson commutation), this paper introduces Jacobi commutation.
  • The contact structure is governed by the Jacobi bracket, a generalization of the Poisson bracket for odd-dimensional manifolds. The condition for commutation is derived from the vanishing of this bracket.

• Approximation and Regularization:

  • To handle non-smooth Hamiltonians or the "almost everywhere" sense of the commutation condition, the author uses mollifiers ($\eta_\delta$) and convex approximations ($f_\epsilon$).
  • A key technique involves analyzing the limit of the difference between composed semi-groups as the time step $\Delta t \to 0$, deriving the necessary algebraic condition on the Hamiltonians.

• Multi-time Equations:

  • The proof strategy links the commutation of semi-groups to the solvability of multi-time Hamilton-Jacobi equations (systems of PDEs with multiple time variables $t_1, \dots, t_d$).

3. Key Contributions and Results

A. The Commutation Condition (H5)

The paper derives a specific algebraic condition, labeled (H5), which is the contact-type analogue of the Poisson commutation condition. For two Hamiltonians $H$ and $F$, they commute if:
$$\begin{aligned} &D_x H D_p F - D_p H D_x F + p \cdot D_p F \cdot \frac{\partial H}{\partial u} - p \cdot D_p H \cdot \frac{\partial F}{\partial u} \\ &+ \frac{\partial F}{\partial u} H - \frac{\partial H}{\partial u} F = 0 \end{aligned}$$
This expression corresponds to the Jacobi bracket $\{H, F\} = 0$.

B. Main Theorems

1. Theorem 2 (Main Result): If $H$ and $F$ satisfy standard growth/convexity assumptions (H1-H4) and the Jacobi commutation condition (H5), their associated Lax-Oleinik semi-groups commute:
   $$S_H(\lambda) \circ S_F(\mu) = S_F(\mu) \circ S_H(\lambda)$$
2. Theorem 3 (Parameter Dependence): It establishes that the solution $v(x, t, \lambda)$ to the equation with Hamiltonian $H + \lambda F$ satisfies a specific evolution equation with respect to the parameter $\lambda$, linking the parameter derivative to the Hamiltonian $F$.
3. Theorem 4 (Multi-time Solvability): Under the commutation condition, the associated multi-time Hamilton-Jacobi system admits a unique viscosity solution. This confirms the equivalence between semi-group commutation and multi-time solvability in the contact setting.
4. Generalizations (Theorems 6, 7, 8):
   • The commutation property is preserved under convex increasing transformations (e.g., $S_H$ and $S_{f(H)}$ commute).
   • The condition (H5) can be relaxed to hold almost everywhere (in the sense of Lebesgue measure) provided the Hamiltonians can be approximated by smooth, convex sequences.
   • The results extend to general closed smooth manifolds (Theorem 9) using the intrinsic Jacobi structure.

C. Inequality for Non-Commuting Cases

Theorem 5 provides a result for the case where the Jacobi bracket is non-positive ($\leq 0$). In this scenario, the semi-groups satisfy an inequality:
$$S_H(\lambda) \circ S_F(\mu) \leq S_F(\mu) \circ S_H(\lambda)$$

4. Significance

• Theoretical Advancement: This is the first work to systematically investigate the commutation properties of contact type Hamilton-Jacobi equations from the perspective of Jacobi brackets. It bridges the gap between contact geometry and the theory of viscosity solutions.
• Geometric Quantization: By identifying the subspace of functions that commute under the Jacobi bracket, the paper contributes to the understanding of geometric quantization on contact manifolds.
• Weak KAM Theory: The results extend the scope of Weak KAM theory (previously focused on Tonelli/Poisson systems) to contact systems, implying that commuting contact Hamiltonians share the same weak KAM solutions, Aubry sets, and Peierls barriers.
• Robustness: The extension of these results to non-smooth Hamiltonians (via approximation) and general manifolds makes the theory applicable to a broader range of physical and geometric problems, including those involving dissipation or non-conservative forces often modeled by contact structures.

In summary, Guyu Jin successfully generalizes the theory of commuting semi-groups to the contact setting, proving that the algebraic condition for commutation is the vanishing of the Jacobi bracket, and demonstrating that this property ensures the solvability of multi-time contact Hamilton-Jacobi systems.