Navier-Stokes-Korteweg System

• Navier-Stokes-Korteweg system is a set of dissipative-dispersive PDEs that incorporate capillarity forces to model compressible viscous fluids and phase transitions.
• It integrates third-order spatial derivatives to capture interfacial dynamics, making it fundamental for analyzing multiphase flow and diffuse interface phenomena.
• Rigorous analysis confirms existence, energy dissipation, and stability properties while structure-preserving numerical schemes enable accurate, robust simulations.

The Navier-Stokes-Korteweg (NSK) system is a class of dissipative-dispersive partial differential equations modeling the dynamics of compressible viscous fluids with capillarity or phase transition effects. It extends the classical compressible Navier-Stokes equations by incorporating third-order spatial derivative terms that represent internal capillarity forces, typically following the Korteweg theory of diffuse interfaces. The system is fundamental in the mathematical and numerical analysis of multiphase flow, liquid-vapor transition, and related phenomena, particularly through its ability to capture interfacial layer dynamics at mesoscopic scales. Its formulation, well-posedness, relaxation limits, and connections to high-order hyperbolic-parabolic models are the subject of intensive ongoing research.

1. Mathematical Formulation and Core Structure

The general form of the one-dimensional compressible Navier-Stokes-Korteweg system in Lagrangian coordinates involves the specific volume $v(x,t)>0$, velocity $u(x,t)$, pressure $p=p(v)$, viscosity coefficient $\mu>0$, and capillarity coefficient $\kappa>0$. The basic system is

$$\begin{cases} v_{t}-u_{x}=0,\ u_{t}+(p(v))_{x} = \mu\left( \frac{u_{x}} {v} \right)_x + \kappa v_{xxx}. \end{cases}$$

The initial data and boundary conditions are prescribed based on the physical setting, with prototypical choices such as $u(0,t)=0$, $v_x(0,t)=0$ to ensure no-flow and no capillarity-flux boundaries (Chen et al., 2017).

For compressible fluids in multi-dimensions, with density $\rho(x,t)\ge0$, velocity $u(x,t)\in\mathbb R^d$, and barotropic pressure law $P(\rho)$, the Eulerian form is

$$\begin{cases} \partial_t\rho + \nabla\cdot(\rho u) = 0, \ \partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla P(\rho) = \nabla\cdot S(\rho,\nabla u) + \nabla\cdot K(\rho), \end{cases}$$

with viscosity and capillarity stress tensors: $$S(\rho,\nabla u)=\mu(\rho) D(u), \qquad D(u)=\tfrac12(\nabla u + \nabla u^T),$$

$$\nabla\cdot K(\rho) = \nabla\Big[ \rho\Delta\rho \Big] - \nabla\cdot (\nabla\rho\otimes\nabla\rho).$$

This structure arises, for instance, with $\mu(\rho)=\rho$ and $\kappa(\rho)=1$ as in the isothermal model on the torus $T^3$ (Antonelli et al., 2019).

2. Existence and Regularity of Solutions

Broad classes of solutions have been rigorously constructed for the Navier-Stokes-Korteweg system:

• Strong Solutions and Global Regularity: For the compressible Dunn-Serrin type NSK system with $\mu(\rho)=\rho$, $\kappa(\rho)=1/\rho$, and polytropic pressure $P(\rho)=\rho^\gamma$, there exist unique global strong solutions on the periodic torus in any dimension $N=2,3$, for arbitrarily large initial data that are strictly positive and regular ($\rho_0 \in H^3$, $u_0 \in H^2$) (Huang et al., 31 Jan 2026). The key technical advance is a logarithmic control on the $L^\infty$ norm of the effective velocity $v = u + \nabla\log \rho$ in terms of the reciprocal density, preventing vacuum formation:

$$\|v\|_{L^\infty} \le C\sqrt{ \ln(e + \|\rho^{-1}\|_{L^\infty}) }.$$

This allows uniform positive lower bounds for the density, closure of higher-order energy estimates, and precludes finite-time blow-up.

• Weak Solutions with Vacuum: Global finite-energy weak solutions exist on $T^3$ for degenerate viscosity $\mu(\rho)=\rho$ and constant capillarity, allowing vacuum regions and large initial data (Antonelli et al., 2019, Antonelli et al., 2018). These solutions satisfy the energy inequality

$$\sup_{t} \int_{T^3} \left( \tfrac12 \rho |u|^2 + F(\rho) + \tfrac12 |\nabla \rho|^2 \right) dx + \int_0^T \int_{T^3} \rho |D u|^2 dx dt \le \text{(data)}$$

and the Bresch–Desjardins (BD) entropy

$$\int_{T^3}|\nabla\sqrt{\rho}|^2(t)dx + \int_0^T \int_{T^3} |\nabla^2\rho|^2 + \rho |\nabla^2\log\rho|^2 dxdt \le C(\rho^0, u^0).$$

Compactness for subsequence convergence foregoes additional damping or BD–Korteweg relation (Antonelli et al., 2018, Antonelli et al., 2019).

• Critical and Non-monotone Pressures: With Van-der-Waals or more general non-monotone $P(\rho)$, global well-posedness in critical Besov spaces is established for small perturbations around constant states, and the parabolic structure persists for $P'(\rho_*) \ge 0$ (Chikami et al., 2018, Takayuki et al., 2019, Kobayashi et al., 2019).
• Incompressible Navier-Stokes-Korteweg: For variable capillarity and viscosity, global existence and uniqueness holds in critical spaces close to equilibrium, utilizing maximal regularity properties and a change of unknowns to represent the system as a coupled heat-Stokes-Lamé type structure (Wang, 2024).

3. Energy, Entropy, and A Priori Structure

NSK systems possess two fundamental dissipative structures:

• The energy dissipation law ensures decay of kinetic, compressible, and capillarity energy:

$$\frac{d}{dt} E(t) + \int \text{dissipation terms} = 0,\qquad E(t) = \int\left[ \tfrac12 \rho |u|^2 + F(\rho) + \tfrac12 |\nabla \rho|^2 \right].$$

• The BD entropy introduces an effective velocity (often $w = u + 2\nu \nabla\log\rho$), providing additional control, especially near vacuum, and crucial compactness for weak convergence (Antonelli et al., 2018, Antonelli et al., 2019).

Many analyses exploit transformations to the effective velocity variables to recover strong coercivity for the energy functional and to uncover uniform parabolicity even in degenerate or singular regimes (Germain et al., 2012, Huang et al., 31 Jan 2026).

4. Relaxation, Nonlocal Models, and Limits

The third-order Korteweg term presents analytic and computational challenges. Recent progress includes the development and rigorous justification of relaxation models that approximate NSK with second-order systems:

• Parabolic Relaxation Systems: Introduction of a relaxation variable $c(x,t)$ and parameters $(\alpha,\beta)>0$ gives a system:

$$\begin{aligned} &\rho_t + \nabla\cdot(\rho u) = 0, \ &(\rho u)_t + \nabla\cdot (\rho u \otimes u + p_\alpha(\rho)I) = \nabla\cdot S(\nabla u) + \alpha \rho \nabla c, \ &c_t - \beta^2 \Delta c = \rho - c, \end{aligned}$$

with $p_\alpha(\rho) = p(\rho) + \frac12\alpha \rho^2$. For $(\alpha\to\infty,\ \beta\to 0)$ one recovers the original high-order system. These systems restore strict hyperbolicity and reduce the need for extended numerical stencils, making them compatible with standard high-order DG or FV solvers (Hitz et al., 2019, Chaudhuri et al., 10 Dec 2025).

• Nonlocal Capillarity Operators: Substitution of the capillarity term by a convolution with a kernel $K_\epsilon$ (nonlocal approximation) enables uniform local well-posedness, and one rigorously demonstrates convergence to the original (local) NSK as the relaxation parameter diverges, and to the standard Navier-Stokes system as capillarity vanishes (Kim et al., 22 Jul 2025).
• High-Friction and Vanishing Capillarity/Viscosity Limits: The high-friction limit and vanishing capillarity/viscosity limits have been rigorously analyzed via relative entropy and effective-dissipation techniques. The resulting equations often correspond to diffusive (Cahn–Hilliard-type) or classical Euler equations (Carnevale et al., 2020, Germain et al., 2012, Burtea et al., 2021).

5. Stability, Periodicity, and Shock Dynamics

• Stability of Viscous Shocks: For the 1D compressible NSK system with appropriate boundary, shock, and mass-balance conditions, viscous shock profiles are shown to be asymptotically stable under small perturbations (Chen et al., 2017). This involves high-order $L^2$-energy estimates, compensation of boundary mass-loss via a phase shift of the shock, and control of spurious boundary integrals through exponential decay.
• Time-Periodic Solutions: For compressible Korteweg-type systems under time-periodic forcing in high dimensions $(n\geq5)$, unique small-amplitude time-periodic solutions exist, with corresponding time-asymptotic stability (Chen et al., 2012). The argument blends decay estimates for the linearized system and energy methods for the nonlinear evolution.
• Diffusion-Wave Asymptotics: The presence of the Korteweg tensor enhances smoothing (parabolic regularity) at all frequencies and gives rise to diffusion-wave phenomena in long-time, low-frequency regimes, with $L^2$ and $L^\infty$ decay rates optimal for heat propagation, even when the linearized pressure derivative vanishes (Kobayashi et al., 2019).

6. Numerical Methods: Structure-Preserving Discretizations

• Finite-Volume and Discontinuous Galerkin Schemes: Recent structure-preserving numerical schemes for the NSK system ensure mass and momentum conservation, exact energy dissipation consistent with the continuous PDE, and allow robust simulation of both regular flows and shocks (Giesselmann et al., 13 Jan 2026, Giesselmann et al., 2012). Key features include:
  • Direct discretization of third-order capillarity terms using symmetric cross-averages.
  • Energy stability via dedicated entropy-variable multipliers and summation-by-parts formulae.
  • Subcell FV/DG stabilization for under-resolved interfaces and shocks.
  • Theoretical proofs and validation for both manufactured and physically relevant test cases.
• Thermodynamically Consistent Relaxation Schemes: Numerical implementation benefits from the relaxation paradigm, which brings capillarity contributions to second order, permitting use of standard upwind fluxes, with strict hyperbolicity ensured through modified pressure laws. For non-isothermal flows, such relaxation approaches also allow the imposition of contact-angle boundary conditions in a thermodynamically consistent way (Keim et al., 2022).

7. Connections, Extensions, and Outlook

• Relative Entropy and Weak-Strong Uniqueness: The augmented-variable (drift velocity/effective velocity) framework enables the establishment of relative-entropy principles both for the original NSK and its relaxation limits. Weak-strong uniqueness holds: weak and strong solutions emanating from the same data coincide as long as the strong solution exists, even with non-monotone pressure laws (Bresch et al., 2017, Chaudhuri et al., 10 Dec 2025).
• Analytic Regularity and Pseudo-measure Frameworks: Analytic smoothing effects for the NSK system with quantum pressure have been established in pseudo-measure spaces, yielding explicit lower bounds on the analyticity radius and new perspectives toward Fourier-frequency based turbulence analysis (Soler, 2022).
• Nonlocal and Phase-Field Generalizations: The nonlocal relaxation approach provides a rigorous multiscale pathway between phase-field models and sharp-interface hydrodynamics, with uniform time intervals of well-posedness and quantitative convergence as nonlocality or capillarity parameters vary (Kim et al., 22 Jul 2025).
• Physical and Thermodynamical Consistency: Proper capillarity modeling, consistent with van der Waals theory and variational free energies, permits the simulation of diffuse interface phenomena, droplet dynamics, and phase transition, with accurate reproduction of sharp and diffuse interface limits, robust handling of contact angles, and monotonic energy dissipation (Hitz et al., 2019, Keim et al., 2022).

The ongoing refinement of well-posedness theories, development of structure-preserving algorithms, and exploration of singular, relaxation, and turbulence-influenced limits continue to drive research on the NSK system and its applications in the mathematical fluid mechanics and multiphase flow community.