Dissipative Contact Hamilton–Jacobi Equation

Updated 5 December 2025

The dissipative contact Hamilton–Jacobi equation is an extension of classical HJ theory that incorporates explicit dissipation through the dependence on the unknown function and contact geometry.
Weak KAM theory and viscosity solution methods provide robust frameworks for establishing existence, uniqueness, and exponential convergence in dynamic nonconservative systems.
Its variational formulation and adaptability to nonlinear, stochastic, and time-periodic extensions underscore its broad applicability in mechanics, optimal control, and dynamical systems.

The dissipative contact Hamilton–Jacobi equation generalizes classical Hamilton–Jacobi theory by incorporating explicit dissipation through dependence on the unknown function itself. This equation arises naturally in the context of contact geometry—where phase space is odd-dimensional and the dynamics generically break symplectic conservation via the Reeb vector field—and encodes a broad class of “dissipative” phenomena. It underpins the analysis of nonconservative mechanical systems, discounted optimal control, and the dynamics of Aubry–Mather sets under weak KAM theory, yielding distinct existence, uniqueness, convergence, and variational characterization results that set it apart from the symplectic case.

1. Contact Hamilton–Jacobi Equation: Geometric and Analytical Formulation

The stationary (time-independent) contact Hamilton–Jacobi equation is given on a closed, connected, smooth Riemannian manifold $M$ as

$H(x, u(x), Du(x)) = 0, \quad x \in M,$

where $H \in C^3(T^*M \times \mathbb{R})$ typically satisfies the Tonelli conditions: strict convexity and superlinearity in $p$ , together with monotonicity or Lipschitz bounds in the "contact variable" $u$ (see (Wang et al., 2018, Liu et al., 22 Sep 2025)). The evolutionary (Cauchy) problem takes the form

$w_t + H(x, w, D_x w) = 0, \quad w(x, 0) = \varphi(x), \quad (x, t) \in M \times (0, +\infty),$

with the dissipative effect arising from terms such as $\partial_u H < 0$ or, more generally, monotonicity in $u$ .

Contact geometry provides the natural geometric underpinning: on a $(2n+1)$ -dimensional manifold $M$ , a contact form $\eta$ yields a unique Reeb vector field $R$ (see Darboux coordinates $(q^i, p_i, z)$ ), and the associated contact Hamiltonian vector field $X_H$ is defined by

$\iota_{X_H} d\eta = dH - (R(H)) \eta, \quad \iota_{X_H} \eta = -H,$

leading to the characteristic ODEs for $(x, p, u)$ (see (León et al., 2021, Esen et al., 2022, Grillo et al., 2019)).

2. Weak KAM/Viscosity Solutions and Comparison Principles

Analysis of stationary and evolutionary contact HJ equations proceeds via weak KAM theory and viscosity solutions. Under strict convexity in $p$ and moderate decrease (or nondecrease) in $u$ , there exist two nonlinear semigroups acting on $C(M)$ : forward and backward solution operators via the implicit Lagrangian variational principle,

$L(x, u, v) := \sup_{p \in T^*_x M} \{ \langle p, v \rangle - H(x, u, p) \}.$

A continuous $u: M \to \mathbb{R}$ is a backward weak KAM solution for $H(x, -u, -Du) = 0$ if it satisfies an action inequality along piecewise $C^1$ curves and admits calibrated characteristics; forward weak KAM solutions are defined analogously.

Comparison principles in the dissipative setting can be significantly more robust than in the conservative case, especially when strict monotonicity in $u$ holds; the solution set is compact and order-bounded in $C(M)$ . In the setting where $H$ is merely nondecreasing in $u$ , refined “generalized comparison principles” utilize the structure of Mather measures, leading to integral orderings and uniqueness criteria tied to "ordinal Mather measures" (see (Liu et al., 22 Sep 2025)).

3. Existence, Uniqueness, and Structure of Solution Sets

Under the Tonelli hypotheses plus “admissibility” (the existence of a critical value for the u-frozen Hamiltonian), compactness and structure of the viscosity solution set $VS(H)$ is guaranteed:

$VS(H)$ admits minimal and maximal elements, $u_-$ and $u_+$ , which correspond uniquely to backward and forward weak KAM solutions, respectively (Wang et al., 2018).
On the Aubry set, $u_-$ and $u_+$ agree and are $C^{1,1}$ regular; outside the Aubry set, the solution set exhibits strict ordering.
Uniqueness may fail if $H$ is only nondecreasing in $u$ , unless ordinal degeneracy is absent for the corresponding Mather measure (Liu et al., 22 Sep 2025).

4. Long-Time Dynamics and Stability

Evolutionary contact HJ equations manifest markedly dissipative long-time behavior. The solution semigroup $T_t \varphi$ exhibits exponential convergence to the maximal stationary solution $u_+$ if the initial data lies in the appropriate order interval: $|T_t \varphi - u_+|_{C^0} = O(e^{-\lambda t}),$ with $\lambda$ determined by uniform bounds on $\partial_u H$ (Jin et al., 2017). Lyapunov and asymptotic stability are characterized through the strict positivity of $\partial_u H$ on the Aubry set, and explicit construction of sub-/super-solutions yields sharp exponential rates (see (Xu et al., 26 Jan 2024, Wu et al., 9 Jan 2025)). The absence of dissipation (i.e. $H$ independent of $u$ ) precludes such uniform rates.

In the presence of perturbations, local Lyapunov asymptotic stability of a viscosity solution $u_-$ persists for the perturbed equation, and the perturbed solution $u_-^\varepsilon$ converges uniformly to $u_-$ as $\varepsilon \to 0$ , with explicit Lipschitz dependence (Wu et al., 9 Jan 2025).

5. Variational Principles and Dynamical Analysis

A defining analytical tool is the "implicit variational principle," encoding solutions as minimizers of nonstandard action functionals reflecting dissipation: $h_{x_0, u_0}(x, t) = u_0 + \inf_{y(0)=x_0, y(t)=x} \int_0^t L(y(s), h(y(s), s), \dot y(s)) \, ds,$ with the underlying action-minimizing orbits interpolating between the Lagrangian and the dissipative contact-Hamiltonian flow.

Forward weak KAM solutions (maximal viscosity solutions) are retrieved via variational formulas involving Peierls barriers and Mather measures, yielding links to discounted optimal control and ergodic problems (Wang et al., 2018, Chen et al., 2018). The discounted case $H(x,u,p) = H_0(x,p) - \alpha u$ recovers classical infinite-horizon formulas via exponential weighting.

6. Contact Geometry, Dissipation, and Integrability

Contact geometry is pivotal both philosophically and technically. The phase space is a $(2n+1)$ -dimensional contact manifold $(M, \eta)$ , where the loss of symplectic volume via the Reeb vector field encodes the dissipative mechanism directly. The contact Hamilton–Jacobi equation,

$H(q, \partial W / \partial q, W(q)) + R_\eta(H) W(q) = 0,$

arises from lifting the Hamiltonian vector field to a Legendrian (or pseudo-isotropic) section; complete solutions ensure integrability by quadratures (Grillo et al., 2019, León et al., 2016, León et al., 2022). Variants involving explicit $z$ (contact coordinate) dependence capture “action reservoir” effects.

The unified mechanics framework (skew-symmetric algebroids with cocycles) shows the contact HJ equation as a natural generalization: twisted differential operators introduce the friction/dissipation term, recovering all known dissipative HJ equations via a single cohomological statement (Balseiro et al., 2010).

7. Extensions: Nonlinear, Stochastic, and Time-Periodic Contact HJ Equations

Beyond classical contact HJ equations, several modern extensions have emerged:

Vanishing contact structure problem: Families $H^\lambda(x,p,u)$ with vanishing dissipative parameter $\lambda \to 0$ select limiting viscosity solutions for the u-independent case, with convergence rates and variational characterizations paralleling Aubry–Mather theory (Chen et al., 2018).
Stochastic contact Hamilton–Jacobi equations: In odd-dimensional stochastic environments, contact SDEs are encoded via stochastic HJ PDEs involving Stratonovich calculus. Structure-preserving numerical schemes based on stochastic Taylor expansions achieve controlled mean-square convergence and preserve contact geometry (Zhan et al., 17 Nov 2024).
Time-periodic contact HJ solutions: The stability exponent $\mu$ governs the existence of accumulating or escaping periodic profiles in the evolutionary equation, with infinitely many distinct time-periodic viscosity solutions arising in the unstable (repelling) regime (Xu et al., 26 Jan 2024).