Vanishing Viscosity Limit in PDE Analysis

• The vanishing viscosity limit is the study of how solutions to viscous PDEs converge to inviscid equations, emphasizing the emergence of boundary layers and singular measures.
• Analytical frameworks span the Navier–Stokes to Euler transition, Hamilton–Jacobi regularizations, and extensions to compressible and microstructured fluid models.
• Recent research quantifies convergence rates and applies selection principles, such as Kato's criteria, to ensure the rigorous justification of the inviscid limit.

The vanishing viscosity limit is a central concept in the analysis of partial differential equations (PDEs) governing continuum phenomena, particularly in fluid dynamics, conservation laws, and nonlinear Hamilton–Jacobi equations. Informally, it describes the asymptotic behavior of solutions to regularized (viscous) PDEs as the viscosity parameter tends to zero, seeking convergence to a solution of the corresponding inviscid (non-viscous) or hyperbolic equation. The limit process is delicate in the presence of boundaries, interfaces, or geometric complexity, as new singular structures—such as boundary layers, vortex sheets, or measure-valued defects—may emerge. Modern research focuses on the analytic and geometric mechanisms that either obstruct or facilitate the rigorous justification of this limit in physically and mathematically relevant settings.

1. Analytical Frameworks for the Vanishing Viscosity Limit

Several prototypical settings for the vanishing viscosity limit are studied in the literature:

• Incompressible and compressible Navier–Stokes to Euler equations: The limit ν→0 for solutions u^ν of the Navier–Stokes equations (either incompressible or compressible, with constant or degenerate viscidities) seeks strong convergence to corresponding Euler solutions under appropriate initial and boundary conditions. The primary difficulty in the bounded domain case is the formation and analysis of boundary layers and associated dissipative structures (Kelliher, 2014, Chen et al., 2016, Schröder et al., 2 Jul 2025).
• Hamilton–Jacobi equations: Here, vanishing viscosity offers a selection principle for viscosity solutions to first-order nonlinear PDEs, especially on domains such as networks or graphs where classical uniqueness may fail. The method is to consider an elliptic or parabolic regularization with a small parameter ε>0 and to pass to the limit as ε→0 (Camilli et al., 2012).
• Nonlinear conservation laws and compressible models: In both scalar and system settings (e.g., barotropic and full compressible flows), vanishing viscosity constructs entropy solutions in the sense of Lax, Kruzkov, and DiPerna–Lions, often via a framework of relative entropy or measure-valued limits (Huang et al., 2010, Basarić, 2019, Chen et al., 2020, Feireisl et al., 2021).
• Problems involving geometry or microstructure: Recent works rigorously analyze the limit in contexts such as domains with porous structure, free boundaries, viscoelastic effects, or fluid models with microstructure (e.g., micropolar flows or networked domains) (Gu et al., 2023, Gu et al., 2022, Lacave et al., 2015, Wang et al., 26 Aug 2025).

The challenge lies in the precise identification of convergence mode, the role of boundary or interface conditions, the fate of energy dissipation, and the emergence (or absence) of singular structures.

2. Boundary Phenomena: Layer Formation and Kato-type Criteria

A recurring theme is the characterization and control of viscous boundary or internal layers in the limit of vanishing viscosity, primarily in the fluid (Navier-Stokes/Euler) context:

• Kato's boundary dissipation criterion: For the incompressible Navier–Stokes system with no-slip boundary data, Tosio Kato's criterion asserts that strong (L²) convergence to Euler holds if and only if the total viscous dissipation inside a layer of thickness proportional to the viscosity vanishes:

$$\lim_{\nu\to0} \,\nu \int_0^T \int_{\Gamma_{c\nu}} |∇u^ν|^2 \,dx\,dt = 0,$$

where Γ_{cν} is an O(ν) strip adjacent to the boundary (Kelliher, 2014). This characterizes the boundary layer as the only possible obstruction to convergence. The physical content is that viscous effects must localize near the boundary to allow inviscid behavior in the bulk.

• Extensions to inhomogeneous and compressible systems: Analogous results generalize Kato's criterion to variable-density flows (Schröder et al., 2 Jul 2025) and compressible Navier-Stokes–Fourier systems, with extra conditions for the vanishing of dissipation in thin boundary neighborhoods or, for no-slip compressible flows, enhanced "Kato-type" criteria involving gradients or certain interpolation inequalities (Feireisl et al., 2021).
• Absence or mitigation of boundary layers under alternative boundary or system structure: For slip or Navier boundary conditions, or for certain viscoelastic and micropolar systems, uniform higher-order estimates are attainable and boundary layer formation is suppressed, e.g., via the regularizing effects of elastic stress or geometric cancellation in the "good unknowns" of the Lagrangian framework (Gu et al., 2023, Gu et al., 2022, Chen et al., 2016, Wang et al., 26 Aug 2025).
• Vortex sheets and singular measures: In boundary-dominated limits, the defect between viscous and inviscid vorticity may accumulate as a measure supported on the boundary, thereby forming a vortex sheet (Kelliher, 2014, Gie et al., 2017). This measure is canonically determined by the slip (tangential velocity) of the limiting Euler flow.

3. Quantitative Rates and Selection Principles

A key analytic question is the quantification of convergence rates and the selection/uniqueness mechanism provided by the vanishing viscosity limit.

• Convergence rates: For 2D vorticity in the Yudovich class, the optimal global-in-time L² rate is $O\bigl((\nu/|\log\nu|)^{\frac12 \exp(-Ct)}\bigr)$, refining the classical diffusive rate $O(\sqrt{\nu t})$ observed for short times (Seis, 2019). For symmetric flows in channels and pipes, rates such as $O(\nu^{1/4})$ (plane channel) or $O(\nu^{4/5})$ (pipe) are achieved using explicit boundary layer correctors (Gie et al., 2017). Inviscid limits to rarefaction waves for compressible flows yield convergence rates as sharp as $O(\varepsilon^{1/6}|\ln\varepsilon|)$ (Li et al., 2019), and in full 3D with vacuum, up to $O(\varepsilon^a|\ln\varepsilon|)$ depending on thermodynamic exponents (Hou et al., 2023).
• Selection among non-unique weak solutions: In non-uniqueness regimes (e.g., for Euler equations with large data or discontinuous initial values), the vanishing viscosity limit acts as a selection principle. In the setting of 3D incompressible flow with initial data admitting infinitely many weak Euler solutions, the weak limit of Leray-Hopf solutions as viscosity vanishes selects the classical shear flow solution, distinguishing it from other admissible but possibly physically irrelevant solutions (Bardos et al., 2012). This property is connected to the "admissible" solution concepts in weak versus dissipative framework.
• Measure-valued limits and weak-strong uniqueness: In compressible flows, vanishing viscosity can yield measure-valued (dissipative) solutions of the Euler equations carrying possible oscillations or concentrations; but as long as a strong Euler solution exists, the measure-valued limit collapses singularities ("weak-strong uniqueness") (Basarić, 2019).

4. The Vanishing Viscosity Limit for Hamilton–Jacobi Equations and Networks

Beyond fluid models, the method is foundational in the theory of viscosity solutions for nonlinear first-order PDEs, notably Hamilton–Jacobi (HJ) equations. On networks (finite graphs of C^∞ curves), the limit process is as follows (Camilli et al., 2012):

• For each edge e_j, one considers a viscous elliptic regularization:

$$-\varepsilon u''_j + H_j(x, u'_j) = 0, \quad x \in e_j,$$

with Kirchhoff-type conditions at transition vertices:

$$\sum_{j \in \mathrm{Inc}(i)} \beta_{ij} u'_j(v_i) = 0, \quad v_i \text{ a transition vertex}.$$

Under conditions of regularity, monotonicity, coercivity, and compatibility on the Hamiltonians, each viscous problem is uniquely solvable in an appropriate space.

• Key a priori estimates (uniform bounds and Lipschitz norms) are established for u^ε. In the limit ε→0, half-relaxed limits yield viscosity sub/supersolutions of the inviscid HJ equations on edges, and the network junction conditions are verified with barrier techniques.
• The result is uniform convergence of the viscous solution to the unique network viscosity solution, which provides both a selection mechanism and a constructive existence proof for the inviscid problem.

5. Physical and Geometric Mechanisms: Suppression or Promotion of Boundary Layers

The interplay between system structure, geometry, and boundary/interface conditions determines whether strong boundary layers arise in the vanishing viscosity limit.

• Elastic stress and viscoelastic coupling: In compressible viscoelastic systems (e.g., neo-Hookean models), the elastic term $\mathrm{div}(\rho F F^T)$ provides additional ellipticity, enhancing control of normal derivatives at the boundary. This effect can prevent formation of boundary layers under no-slip, in contrast to generic compressible Navier-Stokes systems where strong layers are expected (Gu et al., 2023, Gu et al., 2022).
• Micropolar and microstructured fluids: In fluids with micro-rotation (micropolar) variables or angular viscosities, the vanishing viscosity process may lead to an O(1) boundary layer in the micro-rotation (angular velocity) while leaving the primary velocity field unaffected at leading order. Explicit boundary-layer expansions (matched asymptotics) and sharp energy bounds establish the presence and thickness (O(√ε)) of such layers as optimal (Wang et al., 26 Aug 2025).
• Porous media and perforated domains: In fluids traversing domains with vanishingly small rigid inclusions, the limit process recovers inviscid behavior if the small-scale geometry and viscosity jointly meet explicit scaling regimes, quantified by the ratio of hole size to inter-hole distance and the viscosity. The energy norm convergence is conditional on these scales, and the porous material does not affect the macroscopic inviscid flow provided local Reynolds numbers stay finite (Lacave et al., 2015).

6. Open Problems and Extensions

Several fundamental issues remain open or are the subject of active research:

• Unconditional convergence in the presence of boundaries: For the full Navier–Stokes to Euler limit with no-slip boundary data, global L² convergence for all smooth data remains unproven. The possible breakdown of the Prandtl boundary layer description for certain data or domains is a subject of ongoing investigation (Kelliher, 2014).
• Optimal scaling and boundary layer thickness: Determining the precise relationship between viscosity, boundary layer thickness, and convergence rate in more complex or less symmetric geometries, especially beyond Onsager-critical regularity, is an outstanding technical challenge (Chen et al., 2020).
• Generalization to complex fluids and new types of microstructure: The mechanisms by which elasticity, microstructure, or geometrically induced stress regulate or suppress singularity formation in the limit are an active area, with recent work on viscoelastic, micropolar, and networked models (Gu et al., 2023, Wang et al., 26 Aug 2025, Camilli et al., 2012).
• Inviscid selection principles and non-uniqueness: The vanishing viscosity limit, in both classical and measure-valued settings, serves not just as an analytic tool but as a physical selection principle for admissible inviscid solutions in the presence of non-uniqueness, with further implications for turbulence, mixing, and dissipative dynamics (Bardos et al., 2012, Basarić, 2019).
• Extensions to free boundaries and compressible flows with vacuum: The justification of the vanishing viscosity process in the presence of vacuum and moving domains requires advanced techniques in Lagrangian analysis, energy methods adapted to degenerate parabolicity, and boundary trace control (Gu et al., 2022, Chen et al., 2020, Hou et al., 2023).

The vanishing viscosity limit thus remains a central tool and conceptual theme in mathematical analysis of PDEs, linking regularization, selection, and the emergence (or suppression) of singularities in nonlinear evolution.