Vanishing Viscosity Approximation in PDEs

• Vanishing Viscosity Approximation is a singular-perturbation method that replaces inviscid PDEs with a family of viscous problems indexed by a small parameter, selecting physically relevant weak solutions.
• It quantifies convergence rates for both local and nonlocal regularizations in Hamilton–Jacobi equations, conservation laws, and Mean Field Games using integral and adjoint techniques.
• The method underpins practical applications in numerical regularization, boundary layer analysis, and network transmission, ultimately guiding solution selection and stability in complex PDE models.

Vanishing viscosity approximation is a singular-perturbation method in which an inviscid, first-order, or weakly dissipative PDE is replaced by a family of viscous problems indexed by a small parameter and then studied as the viscosity tends to zero. In the literature represented here, the perturbation can be local, as in an added term $\varepsilon \Delta u$; nonlocal, as in $\varepsilon(-\Delta)^s u$; degenerate, as in $(a(x)+\varepsilon)\Delta u$ with $a(x)\ge 0$; or model-specific, as in Brinkman regularization or viscous stress terms in fluid and multiphase systems. The approximation is used to obtain existence for regularized problems, identify the correct limit notion, quantify convergence rates, and, in several settings, select a distinguished inviscid solution such as an entropy solution, a maximal viscosity solution, or a full first-order Mean Field Game (MFG) solution (Biswas, 2012, Mariani et al., 2011, Ciampa et al., 8 Jul 2026).

1. General mechanism and selection principles

A representative Hamilton–Jacobi regularization is

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$

or, in the nonlocal case,

$$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$

For scalar conservation laws, one considers

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$

In MFG, the approximation is applied to the full coupled system, adding $\varepsilon\Delta u^\varepsilon$ to the Hamilton–Jacobi equation and $\varepsilon\Delta m^\varepsilon$ to the forward Fokker–Planck equation (Goffi, 2024, Mariani et al., 2011, Ciampa et al., 8 Jul 2026).

The selection role depends on the model. For scalar conservation laws, the $\Gamma$-limit of the control cost functional is zero exactly on entropy solutions, and the limiting functional is the total variation of the positive part of the entropy production measure (Mariani et al., 2011). In the radially symmetric eikonal problem on $\varepsilon(-\Delta)^s u$0, vanishing viscosity selects the maximal radial viscosity solution

$\varepsilon(-\Delta)^s u$1

and the paper explicitly notes that uniqueness may fail when $\varepsilon(-\Delta)^s u$2 is merely nonnegative and vanishes somewhere (Meng, 17 Aug 2025). For first-order MFGs, vanishing viscosity is described as the principal tool to select a distinguished solution in the inviscid system, which can be ill-posed or nonunique (Ciampa et al., 8 Jul 2026).

Taken together, these works suggest that vanishing viscosity is not a single theorem but a family of approximation principles whose limiting meaning is equation-dependent: entropy admissibility for conservation laws, viscosity-solution selection for HJ-type equations, and full-system selection for coupled forward-backward models.

2. Quantitative theory for Hamilton–Jacobi and HJB equations

For first-order HJ and HJB equations, the recent literature is dominated by convergence-rate questions and by the contrast between local and nonlocal regularizations. In the critical nonlocal case, adding a small $\varepsilon(-\Delta)^s u$3-Laplacian produces $\varepsilon(-\Delta)^s u$4 regularity for the perturbed solution for positive times, and yields the explicit bound

$\varepsilon(-\Delta)^s u$5

for the vanishing-viscosity limit (Biswas, 2012). A later result for semiconcave solutions on $\varepsilon(-\Delta)^s u$6 shows that the half-Laplacian approximation satisfies

$\varepsilon(-\Delta)^s u$7

and explicitly states that the endpoint $\varepsilon(-\Delta)^s u$8 remains open (Goffi, 2024).

For classical Laplacian regularization with convex Hamiltonians, the rate theory has been substantially sharpened. Under uniform convexity and semiconcave data, the sup-norm rate is $\varepsilon(-\Delta)^s u$9 for $(a(x)+\varepsilon)\Delta u$0; under Lipschitz data, the rate is $(a(x)+\varepsilon)\Delta u$1; and for strictly convex Hamiltonians with superquadratic growth, one obtains $(a(x)+\varepsilon)\Delta u$2 with an explicit exponent depending on the growth parameter $(a(x)+\varepsilon)\Delta u$3 (Cirant et al., 21 Feb 2025). In the possibly degenerate viscous case

$(a(x)+\varepsilon)\Delta u$4

the pointwise rate is

$(a(x)+\varepsilon)\Delta u$5

while averaging the error against a smooth probability density removes the logarithmic loss and gives an $(a(x)+\varepsilon)\Delta u$6 bound (Qi, 6 Jul 2026).

The nonconvex and fully nonlinear case is also now quantitative. For

$(a(x)+\varepsilon)\Delta u$7

approximated by

$(a(x)+\varepsilon)\Delta u$8

the convergence rate in $(a(x)+\varepsilon)\Delta u$9 is

$a(x)\ge 0$0

under Hölder assumptions on the solution, initial data, and Hamiltonian (Cecchin et al., 15 Sep 2025).

Setting	Norm	Rate
Uniformly convex HJ, semiconcave data	$a(x)\ge 0$1	$a(x)\ge 0$2, $a(x)\ge 0$3
Uniformly convex HJ, Lipschitz data	$a(x)\ge 0$4	$a(x)\ge 0$5
Half-Laplacian HJ, semiconcave solution	$a(x)\ge 0$6, $a(x)\ge 0$7	$a(x)\ge 0$8
Degenerate viscous HJ	pointwise / averaged	$a(x)\ge 0$9 / $$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$0
Fully nonlinear, non-convex second-order HJ	$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$1	$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$2

Methodologically, this body of work is notable for its reliance on integral and adjoint techniques rather than purely comparison-based arguments. The papers explicitly invoke Evans-type duality, nonlinear adjoint methods, weighted Hessian estimates, entropy estimates for adjoint densities, and sup/inf-convolution regularization (Goffi, 2024, Qi, 6 Jul 2026, Cirant et al., 21 Feb 2025, Cecchin et al., 15 Sep 2025).

3. Mean Field Games and full-system convergence

In time-dependent first-order MFG on the torus, the vanishing-viscosity approximation takes the form

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$3

with terminal and initial conditions

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$4

The 2026 result studies nonlocal coupling $$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$5 regularizing the Hamilton–Jacobi equation and assumes a $$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$6 Hamiltonian with uniform convexity

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$7

Under these assumptions, it proves

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$8

together with the drift estimate

$$\partial_t u^\varepsilon - \varepsilon \Delta u^\varepsilon + H(Du^\varepsilon)=f$$9

and, under additional mild regularity,

$$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$0

The paper states that prior quantitative results covered only the Hamilton–Jacobi component, whereas these estimates provide quantitative convergence for the full MFG system (Ciampa et al., 8 Jul 2026).

A different 2025 line of work treats MFGs in $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$1 with nonlocal and possibly non-separable Hamiltonians, where the diffusivity constant is $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$2. There the value function, player distribution, and gradient satisfy

$$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$3

and the proof combines FBSDE analysis with a PDE stability estimate for HJ equations (Yu et al., 24 May 2025).

These results make the MFG case distinctive within vanishing-viscosity theory. The approximation is not confined to a scalar value equation: the regularization propagates through the coupled HJ/FP structure, and the main analytical issue is transferring rates from the value function to the density dynamics. The papers explicitly identify the combination of nonlocal regularization, duality methods, and transport/Wasserstein stability as the mechanism enabling full-system estimates (Ciampa et al., 8 Jul 2026, Yu et al., 24 May 2025).

4. Conservation laws, entropy, and local singularity structure

For scalar conservation laws, the fractional-viscosity approximation

$$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$4

admits a variational formulation in which the control cost functionals $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$5 are equicoercive and satisfy a $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$6-liminf inequality. The limiting functional is finite only on weak solutions and equals the total variation of the positive part of the entropy production measure; in particular, it vanishes exactly on entropy solutions (Mariani et al., 2011). This provides a precise large-deviations-style interpretation of entropy admissibility.

For systems, the structure of the viscosity matrix becomes decisive. In a $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$7 triangular system with nonlinear viscosity

$$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$8

assuming $$\partial_t u^\varepsilon + \varepsilon(-\Delta)^s u^\varepsilon + H(Du^\varepsilon)=f.$$9 and $$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$0 commute, one obtains a unique global smooth solution with total variation bound

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$1

an $$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$2-stability estimate, and convergence of the vanishing-viscosity family to the unique Liu entropy solution of the inviscid system for small $$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$3 data (Haspot et al., 6 Mar 2025). The paper presents this as an extension of the Bianchini–Bressan theory beyond the constant-viscosity case.

Recent work also resolves the local asymptotic patterns of viscous approximations near singularities. For scalar strictly convex conservation laws, after suitable local rescaling around a shock, a shock interaction, or a shock-formation point, the viscous solutions converge to “eternal solutions” of the viscous equation: a traveling wave along a shock, an eternal merger profile at shock interaction, or a universal viscous Burgers profile at the formation of a new shock (Bressan et al., 30 Apr 2026). In the system case, matched asymptotic expansions near the first singularity yield the sharp strong-norm rate

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$4

together with the Hölder thresholds

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$5

and a universal viscous Burgers inner profile even for systems with degenerate physical viscosity (Anderson et al., 20 Jun 2025).

This local theory shifts the emphasis from mere $$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$6 convergence to the geometry of singularity formation. Vanishing viscosity then becomes a means of resolving shock layers, interaction zones, and universal inner profiles, not only of selecting the global entropy solution.

5. Networks, transmission conditions, and controllability

On finite networks, vanishing viscosity modifies not only the equation on each edge but also the coupling law at the vertices. For HJ equations on a network $$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$7, the viscous problem is

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$8

on the edges, supplemented at transition vertices by the Kirchhoff-type condition

$$\partial_t u + \partial_x f(u) = -\varepsilon (-\partial_{xx})^s u.$$9

The paper proves existence and uniqueness of a classical viscous solution and shows that as $\varepsilon\Delta u^\varepsilon$0 the solutions converge to the unique network viscosity solution of the first-order problem (Camilli et al., 2012). The Kirchhoff condition is presented there as the expression of the effect of viscosity at the nodes.

For linear transport on finite star-shaped networks, the parabolic approximation

$\varepsilon\Delta u^\varepsilon$1

is coupled at the inner node by transmission conditions that do not impose continuity. As $\varepsilon\Delta u^\varepsilon$2, the solutions converge strongly in $\varepsilon\Delta u^\varepsilon$3 to the unique hyperbolic solution, and the limit transmission coefficients are explicitly inherited from the parabolic coupling parameters: $\varepsilon\Delta u^\varepsilon$4 where $\varepsilon\Delta u^\varepsilon$5 is obtained from an M-matrix inversion (Guarguaglini et al., 2020). This is an explicit node-selection mechanism induced by the viscous approximation.

Vanishing viscosity also enters controllability theory. For the one-dimensional wave equation perturbed by fractional Dirichlet Laplacian terms,

$\varepsilon\Delta u^\varepsilon$6

with $\varepsilon\Delta u^\varepsilon$7, $\varepsilon\Delta u^\varepsilon$8, the control family is uniformly bounded in $\varepsilon\Delta u^\varepsilon$9 and every weak limit point as $\varepsilon\Delta m^\varepsilon$0 is a control for the conservative wave equation (Bugariu et al., 2012). Here the vanishing-viscosity problem is not only about convergence of states but about convergence of control mechanisms.

6. Boundary layers, free boundaries, and computational regularization

In fluid mechanics with boundaries, vanishing viscosity is inseparable from boundary-layer analysis. Kato’s 1983 criterion states that convergence of Navier–Stokes to Euler in $\varepsilon\Delta m^\varepsilon$1 is equivalent to the vanishing of viscous dissipation in a boundary layer of width proportional to the viscosity: $\varepsilon\Delta m^\varepsilon$2 Kelliher’s later result gives an equivalent vorticity formulation and identifies the limit with boundary vortex-sheet formation. In two dimensions, the paper gives the explicit necessary and sufficient criterion

$\varepsilon\Delta m^\varepsilon$3

for the vanishing-viscosity limit (Kelliher, 2014).

For symmetric bounded flows, explicit correctors resolve the boundary layer quantitatively. In plane-parallel channel flows and parallel pipe flows, one constructs boundary layer correctors $\varepsilon\Delta m^\varepsilon$4 so that

$\varepsilon\Delta m^\varepsilon$5

and proves

$\varepsilon\Delta m^\varepsilon$6

The same analysis shows that vorticity accumulates as a vortex sheet on the boundary in the vanishing-viscosity limit (Gie et al., 2017).

The method also appears in compressible and free-boundary problems. For the two-dimensional compressible isentropic Navier–Stokes equations converging to a planar rarefaction wave of Euler, the introduction of a hyperbolic wave is described as crucial to recover the physical viscosities of the inviscid rarefaction profile, and the convergence rate away from the initial time is

$\varepsilon\Delta m^\varepsilon$7

(Li et al., 2019). In a nonlinear tumor-growth model with moving domain, Brinkman regularization

$\varepsilon\Delta m^\varepsilon$8

provides the compactness and regularity needed to construct global-in-time weak solutions and then pass to the limit $\varepsilon\Delta m^\varepsilon$9 (Donatelli et al., 2017).

The computational counterpart is explicit in numerical Euler approximation. The Viscosity Finite Volume method computes numerical solutions of viscous approximants and then uses S-convergence, based on averaging, to obtain a viscosity solution of the Euler system as the S-limit of the numerical family (Feireisl et al., 2021). A plausible implication is that, in practice, vanishing viscosity is simultaneously an analytical selection device and a numerical regularization principle: it identifies meaningful weak or measure-valued limits while supplying explicit error bounds or stability criteria in regimes where direct inviscid analysis is substantially more singular.