Metric Viscosity Solutions

• Metric viscosity solutions are weak formulations that replace classical gradients with metric slopes or curvewise moduli in non-smooth spaces.
• They encompass various frameworks, including curve-based, slope-based, and Monge formulations in Hamilton–Jacobi equations across geodesic, Wasserstein, and rate-independent settings.
• These approaches provide robust stability, comparison principles, and variational characterizations essential for analysis in abstract metric and geometric spaces.

Searching arXiv for recent and foundational papers on metric viscosity solutions across metric-space Hamilton–Jacobi theory, Wasserstein-space PDEs, and geometric extensions. Metric viscosity solutions are weak solution concepts that transplant viscosity-theoretic reasoning from smooth Euclidean settings to spaces where the ambient structure is metric rather than linear. In the first-order Hamilton–Jacobi literature, the basic replacement for the classical gradient is a metric slope or a curvewise modulus of the gradient, and the viscosity inequalities are formulated through admissible curves, local slopes, or distance-based test functions on general metric, length, or geodesic spaces (Nakayasu, 2014). In a broader geometric usage, the same label also covers intrinsic viscosity theories on Wasserstein spaces and viscosity-type formulations for rate-independent systems in abstract metric spaces, where “viscosity” is encoded through localized stability, penalized incremental minimization, and metric transition costs rather than through smooth differential operators (Cheung et al., 2023, Rossi et al., 2017). The subject therefore does not consist of a single universal definition; it is a family of related frameworks whose common feature is that the metric structure replaces the classical differential one.

1. Conceptual range and principal settings

The modern literature uses “metric viscosity solution” in several technically distinct but structurally related ways. The common template is an equation or evolution law posed on a space without a usable linear tangent structure, together with a notion of first- or second-order information expressed through the metric, through absolutely continuous curves, or through distance-based penalizations.

Framework	Ambient space	Defining mechanism
Curve-based Hamilton–Jacobi theory	General metric or length spaces	Admissible curves, metric derivative, DPP
Slope-based Hamilton–Jacobi theory	Length or geodesic spaces	Local slopes and metric test functions
Monge solutions	Complete length spaces	Pointwise slope/Lagrangian identities
Wasserstein HJB theory	$\mathcal{P}_2(\mathbb{R}^d)$	Lions derivatives, $\mathcal{H}$, smooth metric gauges
Rate-independent systems	Complete metric spaces	Stability plus augmented variation and jump costs

For first-order Hamilton–Jacobi equations, a prototypical model is

$u_t + H(x,|Du|)=0,$

posed on a metric space $(X,d)$, with $H(x,p)$ depending only on the modulus of the spatial gradient (Nakayasu, 2014). In that setting, “metric viscosity” refers to replacing $Du$ by curvewise or slope-based quantities. For the eikonal equation, the metric formulation is particularly tight: on complete length spaces, 1-Lipschitz solutions of $|du|=1$ are characterized by a pointwise metric slope condition and are exactly distance-like functions in the global setting (Jiang et al., 2023).

A broader extension occurs on the Wasserstein space, where the underlying topology is still that of $\mathcal{W}_2$, but comparison proofs for second-order HJB equations use an auxiliary smooth metric $SW_2^\sigma$ and gauge functions built from it (Cheung et al., 2023). Another extension appears in the theory of rate-independent systems, where Visco-Energetic and Balanced Viscosity evolutions are described as metric viscosity solutions because viscosity effects are encoded by quadratic metric penalties, localized stability, and metric jump costs (Rossi et al., 2017).

This dispersion of usages suggests that “metric viscosity solution” is best understood as an umbrella term for intrinsically metric weak formulations rather than as a single canonical definition.

2. First-order Hamilton–Jacobi equations on metric spaces

A foundational metric-space theory is developed for evolution equations on arbitrary metric spaces $(X,d)$, without assuming linear structure and, in one formulation, without restricting to geodesic spaces (Nakayasu, 2014). The basic dynamical objects are absolutely continuous curves $\mathcal{H}$0, their metric derivatives

$\mathcal{H}$1

and the Lagrangian $\mathcal{H}$2 obtained from the Hamiltonian by Legendre–Fenchel duality. Under continuity, convexity, and coercivity assumptions on $\mathcal{H}$3, the value function is represented by a variational formula

$\mathcal{H}$4

and this value function is the unique solution of the metric initial-value problem (Nakayasu, 2014).

In the dynamic-programming formulation, a subsolution is characterized by testing $\mathcal{H}$5 along every admissible backward path and requiring a one-dimensional viscosity inequality involving the running cost $\mathcal{H}$6. The supersolution notion is asymmetric: for every $\mathcal{H}$7 and $\mathcal{H}$8, one must find an admissible curve and an auxiliary function $\mathcal{H}$9 that stays above the pullback of $u_t + H(x,|Du|)=0,$0 up to $u_t + H(x,|Du|)=0,$1, while satisfying the reverse one-dimensional inequality. These pathwise inequalities are equivalent to suboptimality and superoptimality principles derived from the DPP (Nakayasu, 2014).

A more Crandall–Lions-like formulation uses metric slopes. In the Gangbo–Święch framework, one defines local slopes, upper slopes, and lower slopes through

$u_t + H(x,|Du|)=0,$2

and similarly for one-sided variants. Test functions are decomposed as $u_t + H(x,|Du|)=0,$3, where $u_t + H(x,|Du|)=0,$4 has a controlled continuous slope and $u_t + H(x,|Du|)=0,$5 is merely locally Lipschitz; the Hamiltonian is correspondingly regularized to absorb the slope contribution of $u_t + H(x,|Du|)=0,$6 (Nakayasu et al., 2016). This yields a viscosity definition that is intrinsic to the metric and supports comparison, Perron-type constructions, and large-time asymptotics on compact geodesic spaces (Nakayasu et al., 2016).

A further simplification, crucial for perturbation theory, is that on locally compact geodesic spaces the class of metric test functions can be reduced to quadratic distance functions. For stationary equations

$u_t + H(x,|Du|)=0,$7

metric viscosity sub- and supersolutions can be characterized using only tests of the form $u_t + H(x,|Du|)=0,$8 at strict local extrema (Makida et al., 2023). This reduction is technically decisive because convergence of such test functions is governed directly by convergence of the underlying metrics.

3. Equivalent formulations and the eikonal archetype

A major development in the subject is the demonstration that several apparently different notions are equivalent under appropriate structural assumptions. In complete rectifiably connected or complete length spaces, the eikonal equation

$u_t + H(x,|Du|)=0,$9

admits three principal formulations: curve-based viscosity solutions, slope-based viscosity solutions, and Monge solutions (Liu et al., 2020).

The curve-based definition tests the equation along admissible curves of speed at most one. The slope-based definition uses local slopes and metric test functions. The Monge definition is pointwise and test-function-free: for a locally Lipschitz $(X,d)$0, the equation is encoded directly by the lower slope, so that in the eikonal case a Monge solution satisfies

$(X,d)$1

On complete length spaces, with $(X,d)$2 and suitable continuity assumptions, these three formulations are locally equivalent, and any such solution is locally Lipschitz with

$(X,d)$3

throughout the domain (Liu et al., 2020).

For more general Hamilton–Jacobi equations $(X,d)$4, the same equivalence persists when $(X,d)$5 is continuous, coercive, and strictly increasing or monotone in the slope variable. The reduction proceeds by defining a pointwise effective eikonal coefficient

$(X,d)$6

so that the original equation becomes equivalent to $(X,d)$7 for the candidate solution. Under these hypotheses, Monge and slope-based solutions coincide; in proper geodesic spaces, the local uniform continuity requirements can be weakened (Liu et al., 2020).

For time-dependent Hamilton–Jacobi equations, a related equivalence is established through the introduction of Monge solutions on complete length spaces. The key step is to rewrite the time-dependent problem as a stationary one on space-time under a one-sided Lipschitz-in-time condition $(X,d)$8, then define a space-time subslope $(X,d)$9 or a Lagrangian space-time slope $H(x,p)$0 for $H(x,p)$1 (Liu et al., 7 Feb 2025). Under convex, nondecreasing, coercive Hamiltonians, bounded Lipschitz Monge solutions exist and are unique, and for time-independent superlinear Hamiltonians they are equivalent both to curve-based metric viscosity solutions and to slope-based metric viscosity solutions (Liu et al., 7 Feb 2025).

The eikonal equation remains the cleanest model. On a complete length space, a 1-Lipschitz function is a metric viscosity solution of $H(x,p)$2 exactly when its local slope satisfies

$H(x,p)$3

and this is equivalent to a dynamic metric formula

$H(x,p)$4

for some $H(x,p)$5 at every point (Jiang et al., 2023). In an unbounded complete length space, this is further equivalent to being a distance-like function. On complete non-compact Riemannian manifolds, the same metric notion coincides with the classical viscosity solution of $H(x,p)$6 (Jiang et al., 2023).

The equivalence theory also clarifies a frequent misconception. The various metric formulations are not automatically interchangeable. For general Hamiltonians, monotonicity in the slope variable is essential; without it, explicit counterexamples show that Monge and slope-based solutions can differ (Liu et al., 2020). Likewise, some stability properties fail under mere pointwise convergence in non-locally compact settings, whereas locally uniform convergence restores stability (Jiang et al., 2023).

4. Wasserstein-space and geometric extensions

A distinct branch of the theory treats Hamilton–Jacobi equations on Wasserstein spaces. For second-order HJB equations associated with mean field control with common noise, the state space is $H(x,p)$7 endowed with $H(x,p)$8, and the value function satisfies a PDE involving Lions derivatives $H(x,p)$9, the mixed derivative $Du$0, and a weak second-order operator

$Du$1

The viscosity theory is formulated intrinsically on Wasserstein space, but comparison requires a smooth variational principle based on the Gaussian-regularized sliced Wasserstein metric $Du$2 and the gauge

$Du$3

(Cheung et al., 2023).

The test class is $Du$4, which retains $Du$5, $Du$6, $Du$7, and $Du$8 but avoids full second-order Lions smoothness. A central analytical input is an Itô formula on $Du$9 for functions in $|du|=1$0. The resulting subsolution notion resembles Crandall–Lions theory, but the supersolution notion is strengthened: tests are allowed on $|du|=1$1, so the test function may depend on the joint law of state and control, and the minimum is taken globally on that larger space (Cheung et al., 2023). This strengthened supersolution condition is explicitly stated to be stronger than the classical Crandall–Lions formulation and is essential for the comparison argument.

A first-order Wasserstein theory appears in the study of the eikonal equation on $|du|=1$2, where $|du|=1$3 is complete, separable, locally compact, non-compact, and geodesic. There the local slope on Wasserstein space is

$|du|=1$4

and 1-Lipschitz solutions of $|du|=1$5 are metric viscosity solutions in exactly the same sense as on ordinary length spaces (Jiang et al., 2023). Two explicit constructions of strong metric viscosity solutions are given: dlc-functions induced by closed sets satisfying a compactness-type (CS) condition, and integral lifts

$|du|=1$6

of strong metric viscosity solutions on the base space $|du|=1$7 (Jiang et al., 2023).

A broader geometric use of the term appears in complex Hessian equations on bounded domains in $|du|=1$8 and on compact Hermitian homogeneous manifolds. There the operator is built from the Hermitian metric through forms such as

$|du|=1$9

and viscosity sub- and supersolutions are defined through $\mathcal{W}_2$0 test functions touching from above or below, with ellipticity restricted to the cone of $\mathcal{W}_2$1-positive forms (Chinh, 2012). This is not a metric-space Hamilton–Jacobi theory, but it is a geometric realization of viscosity theory in which the ambient Hermitian metric determines the nonlinear operator and the admissible ellipticity cone. A plausible implication is that “metric viscosity” has acquired a broader geometric meaning in parts of the literature, extending beyond purely distance-based first-order equations.

5. Stability, comparison, and asymptotic behavior

Stability theory is one of the main organizing principles of metric viscosity solutions. In the Gangbo–Święch framework on compact geodesic spaces, upper semilimits of subsolutions and lower semilimits of supersolutions remain sub- and supersolutions under a local compactness assumption on the underlying space; this yields stability under extremal envelopes and supports a Perron method for existence (Nakayasu et al., 2016). On compact geodesic spaces, comparison for

$\mathcal{W}_2$2

leads to uniqueness, and under convexity and coercivity assumptions on $\mathcal{W}_2$3, the large-time asymptotics are governed by the ergodic constant

$\mathcal{W}_2$4

with

$\mathcal{W}_2$5

locally uniformly, where $\mathcal{W}_2$6 solves the stationary ergodic problem $\mathcal{W}_2$7 (Nakayasu et al., 2016).

When the underlying spaces vary, stability requires the metric structure itself to converge. Under Hausdorff convergence of compact subsets $\mathcal{W}_2$8, convergence of the intrinsic metrics $\mathcal{W}_2$9, and convergence of Hamiltonians $SW_2^\sigma$0, half-relaxed limits of metric viscosity sub- and supersolutions solve the limit equation on $SW_2^\sigma$1 (Makida et al., 2023). The argument depends on the reduction of admissible test functions to squared distances $SW_2^\sigma$2, because the slopes of these tests are explicit and stable under metric convergence. The theory covers network approximations of the Vicsek fractal and the Sierpiński gasket, shrinking tube junctions converging to graph-like junctions, and lattice lines converging to the Manhattan plane (Makida et al., 2023).

A more intrinsic convergence result is stability under Gromov–Hausdorff convergence. For first-order Hamilton–Jacobi equations on locally compact geodesic spaces,

$SW_2^\sigma$3

metric viscosity solutions are shown to be stable under Gromov–Hausdorff convergence using $SW_2^\sigma$4-isometries rather than embeddings into a common ambient space (Makida, 9 Jul 2025). The proof combines a characterization of sub- and supersolutions via quadratic distance functions

$SW_2^\sigma$5

with a doubling-variable method adapted to the $SW_2^\sigma$6-isometries. As a byproduct, the same PDE machinery yields a proof of stability for maximizers in the dual Kantorovich problem under measured-Gromov–Hausdorff convergence (Makida, 9 Jul 2025).

Stability is not purely formal. In the Wasserstein eikonal theory, pointwise convergence of strong metric viscosity solutions can fail to preserve the solution property on non-locally compact spaces; locally uniform convergence is the stable mode (Jiang et al., 2023). This is a recurring theme: metric viscosity theories retain the half-relaxed-limit logic of classical viscosity theory, but the compactness mechanism depends much more sensitively on the geometry of the underlying space.

6. Metric viscosity in abstract rate-independent systems

A second major usage of “metric viscosity solutions” arises in the theory of rate-independent systems on complete metric spaces $SW_2^\sigma$7. The state space carries a time-dependent energy

$SW_2^\sigma$8

and the dissipation is the metric distance $SW_2^\sigma$9. Since solutions are only of bounded variation, the evolution is expressed through stability conditions and energy balances rather than through a pointwise differential inclusion (Rossi et al., 2017).

Three notions are central. An Energetic solution satisfies global stability

$(X,d)$0

and an energy balance involving the total variation $(X,d)$1. A Balanced Viscosity solution satisfies only local stability off the jump set,

$(X,d)$2

and an energy balance in which jumps are charged by a viscous transition cost $(X,d)$3 computed by minimizing over absolutely continuous transition curves (Rossi et al., 2017).

Visco-Energetic solutions interpolate between these two. They arise from the incremental scheme

$(X,d)$4

which adds a fixed quadratic metric correction. The induced stability notion is $(X,d)$5-stability,

$(X,d)$6

and the deviation from stability is measured by the residual

$(X,d)$7

Jumps are resolved by a metric transition cost $(X,d)$8 that combines ordinary metric variation, quadratic gap penalties across holes in a compact transition set, and cumulative residual-stability penalties (Rossi et al., 2017).

In this setting, “viscosity” is not the Crandall–Lions notion. It refers to a metric regularization mechanism: a fixed or vanishing quadratic penalty localizes stability and selects jump trajectories. The singular limits of the Visco-Energetic parameter make this interpretation precise. As $(X,d)$9, Visco-Energetic solutions converge to Energetic solutions; as $\mathcal{H}$00, they converge to Balanced Viscosity solutions (Rossi et al., 2017). This places Visco-Energetic evolution between global energetic selection and fully viscous jump resolution, and it motivates the description of VE and BV solutions as metric viscosity solutions for rate-independent systems.

A common misconception is that all metric viscosity theories concern first-order Hamilton–Jacobi equations. The rate-independent literature shows a parallel but distinct use: viscosity is encoded through penalized incremental minimization, localized stability, and augmented jump variation rather than through test functions and metric slopes.

7. Structural themes and interpretive synthesis

Across these theories, several recurrent structures define the metric viscosity paradigm. First, the metric replaces the classical differential structure: gradients become local slopes, metric derivatives along curves, or distance-based penalization terms. Second, variational or dynamic-programming principles frequently underlie the definition, even when the final viscosity notion is stated through local tests. Third, comparison and stability are decisive: metric viscosity notions are typically judged by whether they support DPP characterizations, comparison theorems, Perron constructions, and robust convergence under perturbations of data or geometry (Nakayasu, 2014, Nakayasu et al., 2016, Makida, 9 Jul 2025).

At the same time, the literature makes clear that the term is not monolithic. In first-order Hamilton–Jacobi theory, curve-based, slope-based, and Monge formulations can be equivalent, but only under specific hypotheses on the space and the Hamiltonian (Liu et al., 2020, Liu et al., 7 Feb 2025). In Wasserstein second-order HJB theory, the supersolution concept may need to be strengthened beyond the standard Crandall–Lions template to obtain comparison (Cheung et al., 2023). In rate-independent systems, the same phrase denotes a viscosity-type selection mechanism for jumps rather than a slope-test definition (Rossi et al., 2017).

This suggests a precise broad view. Metric viscosity solutions are weak solutions whose defining inequalities, selection principles, or jump costs are expressed intrinsically in terms of the metric geometry of the state space. In the most classical branch, this means Hamilton–Jacobi equations on general metric spaces. In a wider sense, it includes geometric and variational theories where viscosity effects are encoded through metric slopes, smooth metric gauges, or metricized dissipation mechanisms.