Navier-Stokes Equations: Definition & Applications

• Navier-Stokes system describes viscous fluid motion both compressible and incompressible with velocity $\mathbf{V}$ and pressure, $p$.
• Nonlinear term $\mathbf{V} \cdot \nabla \mathbf{V}$ complicates the system and makes three-dimensional global regularity still unsolved. The convective term produces quadratic and cubic
• Variables and forcing terms all create and control turbulence, as well as shear stresses in fluids.
• mechanically unstable flows to address problems of long term uncertainties.

The Navier–Stokes system is a set of nonlinear partial differential equations describing the motion of viscous fluids. In its incompressible Newtonian form, the unknowns are the velocity field $\mathbf V(\mathbf r,t)$ and pressure $p(\mathbf r,t)$, with constant density $\rho_0>0$ and kinematic viscosity $\nu>0$:

$\nabla\cdot\mathbf V=0,$

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$

The first equation expresses incompressibility; the second balances material acceleration against pressure, viscous diffusion, and external force. The system is nonlinear through the convective term $\mathbf V\cdot\nabla\mathbf V$, and the three-dimensional global regularity problem for arbitrary smooth data remains unresolved. Navier–Stokes equations also occur in compressible, thermal, stochastic, kinetic, geometric, variational, and multiphase formulations.

1. Governing equations and physical structure

For an incompressible Newtonian fluid, the mass density is fixed:

$\rho(\mathbf r,t)=\rho_0.$

The material derivative is

$$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$

and the momentum equation can be written

$$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$

where

$p(\mathbf r,t)$0

More generally, if the volume force density is

$p(\mathbf r,t)$1

the reduced pressure

$p(\mathbf r,t)$2

absorbs the potential contribution, giving

$p(\mathbf r,t)$3

Taking the divergence of the momentum equation and using $p(\mathbf r,t)$4 yields the pressure Poisson equation

$p(\mathbf r,t)$5

Thus pressure is determined elliptically from the velocity and forcing, rather than evolving through an independent parabolic equation.

The standard initial-boundary-value problem specifies

$p(\mathbf r,t)$6

and boundary data on $p(\mathbf r,t)$7. On bounded domains, the no-slip condition is commonly written

$p(\mathbf r,t)$8

For whole-space problems, $p(\mathbf r,t)$9 and physical boundary conditions are absent. Periodic domains replace spatial boundary conditions by periodicity.

The energy identity for the unforced incompressible system is formally

$\rho_0>0$0

The convective and pressure terms do no work under incompressibility and suitable boundary conditions. Viscosity therefore dissipates kinetic energy. On a periodic domain with mean-zero velocity, Poincaré’s inequality implies exponential energy decay for the exact unforced solution (Novikov et al., 2017).

2. Mathematical formulations and pressure elimination

The pressure can be eliminated by the Leray projection

$\rho_0>0$1

which projects vector fields onto divergence-free fields. The projected equation is

$\rho_0>0$2

In weak formulations, the velocity belongs to divergence-free function spaces such as

$\rho_0>0$3

and

$\rho_0>0$4

The weak equation is expressed through

$\rho_0>0$5

A Leray–Hopf weak solution satisfies

$\rho_0>0$6

with a time derivative in an appropriate dual space and the energy inequality

$\rho_0>0$7

A variational WIDE construction obtains such a solution as a subsequential limit of minimizers of an elliptic-in-time functional. The functional contains the squared material acceleration, viscous dissipation, and stabilization terms weighted by $\rho_0>0$8. The regularized Euler–Lagrange equation contains the Navier–Stokes operator plus terms multiplied by $\rho_0>0$9, which vanish as $\nu>0$0 (Ortiz et al., 2018).

A separate least-squares formulation introduces a parabolic corrector $\nu>0$1 satisfying

$\nu>0$2

and minimizes

$\nu>0$3

The functional vanishes precisely at a Navier–Stokes solution. In two dimensions, a damped Newton iteration converges globally in the analytical sense from arbitrary admissible initial guesses, with quadratic convergence after a finite number of iterations. In three dimensions, the corresponding result is local in time for regular solutions (Lemoine et al., 2019).

3. Existence, regularity, and singular behavior

In two spatial dimensions, the classical cancellation of vortex stretching supports global regularity for broad classes of data. Forward-backward stochastic differential systems provide a probabilistic representation of local strong solutions, and global solutions in the paper’s setting when either the dimension is two or the Reynolds number is sufficiently small (Delbaen et al., 2013).

In three dimensions, local wellposedness is available in critical-type spaces under suitable assumptions. For the Navier–Stokes–Maxwell system, local mild solutions exist for arbitrarily large data in critical-type spaces, while global solutions follow for sufficiently small data. The electromagnetic coupling introduces the Lorentz force

$\nu>0$4

and produces quadratic and cubic nonlinearities. The associated energy identity contains viscous dissipation of the velocity and resistive dissipation of the electric field, while the magnetic field is governed by a damped hyperbolic evolution (Germain et al., 2012).

Global weak existence has also been established for coupled systems. For the Navier–Stokes–BGK model on $\nu>0$5, a nonnegative kinetic distribution $\nu>0$6 is coupled to an incompressible velocity $\nu>0$7 by drag:

$\nu>0$8

$\nu>0$9

The total-energy inequality combines particle kinetic energy, fluid kinetic energy, viscous dissipation, drag dissipation, and BGK entropy dissipation. Existence relies on Dunford–Pettis compactness, velocity averaging, high-order velocity moments, and the Aubin–Lions lemma (Choi et al., 2018).

The full compressible Navier–Stokes–Fourier system includes density $\nabla\cdot\mathbf V=0,$0, velocity $\nabla\cdot\mathbf V=0,$1, and temperature $\nabla\cdot\mathbf V=0,$2:

$\nabla\cdot\mathbf V=0,$3

$\nabla\cdot\mathbf V=0,$4

$\nabla\cdot\mathbf V=0,$5

Here $\nabla\cdot\mathbf V=0,$6 is the Newtonian stress and $\nabla\cdot\mathbf V=0,$7. Global weak solutions are constructed using entropy inequalities, total or ballistic energy balances, artificial pressure and viscosity, Galerkin approximation, effective viscous flux, renormalized continuity equations, and stochastic or deterministic compactness methods (Breit et al., 2017, Feireisl et al., 2020, Chaudhuri et al., 2021).

Special explicit solutions do not address the general regularity problem. Carefully selected spatially periodic divergence-free data can make the pressure projection exactly balance the nonlinear term, reducing the velocity evolution to a heat equation. Such constructions yield smooth periodic solutions in $\nabla\cdot\mathbf V=0,$8 and $\nabla\cdot\mathbf V=0,$9, but only for special data satisfying a strong integral compatibility condition (Thambynayagam, 2022).

Singular forced solutions also exist. For a Hölder-continuous curve $$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$0 in $$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$1, one can construct solutions whose leading behavior is a translated Slezkin–Landau profile:

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$2

For Hölder exponent $$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$3, the solution is smooth outside the space-time curve

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$4

while globally it solves a Navier–Stokes equation with a force concentrated on the curve. This construction does not produce a singular solution of the unforced three-dimensional Cauchy problem (Karch et al., 2014).

4. Alternative dynamical, probabilistic, and kinetic representations

An inverse kinetic theory represents incompressible Navier–Stokes fields as velocity moments of a one-point phase-space probability density $$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$5. The correspondence conditions include

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$6

and

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$7

The PDF satisfies a Liouville equation

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$8

The associated characteristics form a six-dimensional phase-space system:

$$\frac{\partial\mathbf V}{\partial t}+\mathbf V\cdot\nabla\mathbf V =-\frac{1}{\rho_0}\nabla p+\mathbf f_R+\nu\nabla^2\mathbf V.$$9

Under moment correspondence, Liouville evolution, and the requirement that a local Maxwellian be a solution if and only if the macroscopic fields solve strong incompressible Navier–Stokes, the resulting finite-dimensional flow is formally equivalent to the strong stochastic INSE problem. This formal equivalence does not establish global regularity or uniqueness for arbitrary three-dimensional data (Tessarotto et al., 2010).

Probabilistic Lagrangian representations use stochastic flows

$\mathbf V\cdot\nabla\mathbf V$0

and represent the velocity by

$\mathbf V\cdot\nabla\mathbf V$1

A finite-$\mathbf V\cdot\nabla\mathbf V$2 particle approximation replaces the expectation by an empirical mean over independent stochastic flows. The resulting finite-particle system does not automatically reproduce the long-time energy dissipation of the exact equation because dissipation depends on cross-correlations between particle gradients. A resetting procedure periodically replaces the individual particle fields by their empirical mean; in two dimensions this restores energy decay under the stated correlation criterion (Novikov et al., 2017).

Forward-backward stochastic differential systems provide another probabilistic representation. A forward process is driven by Brownian motion, while a backward process represents velocity and pressure. An infinite-horizon backward equation realizes the nonlocal pressure operator

$\mathbf V\cdot\nabla\mathbf V$3

Under Sobolev assumptions, the resulting FBSDS corresponds locally to a strong Navier–Stokes solution. The same formulation connects to stochastic Lagrangian representations associated with Constantin–Iyer and Zhang (Delbaen et al., 2013).

5. Generalizations and coupled systems

Geometric Navier–Stokes equations

On a compact boundaryless Riemannian manifold $\mathbf V\cdot\nabla\mathbf V$4, the incompressible system is

$\mathbf V\cdot\nabla\mathbf V$5

The stress-based diffusion operator is

$\mathbf V\cdot\nabla\mathbf V$6

where

$\mathbf V\cdot\nabla\mathbf V$7

is the symmetric deformation tensor. For divergence-free fields,

$\mathbf V\cdot\nabla\mathbf V$8

where $\mathbf V\cdot\nabla\mathbf V$9 is the Bochner Laplacian and $\rho(\mathbf r,t)=\rho_0.$0 is the Ricci operator. The Hodge Laplacian instead satisfies

$\rho(\mathbf r,t)=\rho_0.$1

The stress-based operator is associated with Newtonian constitutive laws because viscous stress depends on the symmetric rate of deformation. Killing vector fields satisfy $\rho(\mathbf r,t)=\rho_0.$2 and are stationary solutions with

$\rho(\mathbf r,t)=\rho_0.$3

The $\rho(\mathbf r,t)=\rho_0.$4 projection onto Killing fields is conserved, while the orthogonal component decays exponentially under the stated coercivity estimate. On the sphere, this selects rigid rotational modes as persistent unforced motions (Samavaki et al., 2018).

Stochastic transport

Compressible Navier–Stokes equations can be perturbed by transport noise in both continuity and momentum equations:

$\rho(\mathbf r,t)=\rho_0.$5

$\rho(\mathbf r,t)=\rho_0.$6

For solenoidal transport fields,

$\rho(\mathbf r,t)=\rho_0.$7

the stochastic contribution is a spatial flux, and the total energy remains formally compatible with viscous dissipation. Weak solutions exist for smooth signals, geometric rough paths, and Brownian Stratonovich noise under the paper’s assumptions. Strong density convergence remains the central analytical difficulty, and rough or Brownian theories restrict the transport fields to spatially constant vectors (Breit et al., 2021).

Reversible constrained dynamics

A reversible model replaces constant viscosity $\rho(\mathbf r,t)=\rho_0.$8 by a state-dependent multiplier $\rho(\mathbf r,t)=\rho_0.$9 chosen to preserve total enstrophy:

$$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$0

For periodic flows,

$$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$1

The multiplier can become negative, producing temporary anti-dissipative behavior. Numerical experiments for forced homogeneous isotropic turbulence at Taylor-scale Reynolds numbers approximately $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$2, $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$3, and $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$4 found close agreement between ordinary and reversible systems in velocity moments, spectra, temporal correlations, vorticity geometry, and coarse-grained energy fluxes when mean enstrophy and mean effective viscosity are matched. These results are numerical evidence for Gallavotti’s equivalence conjecture, not a theorem about Navier–Stokes dynamics (Jaccod et al., 2020).

Multiphase and thermodynamic coupling

Nonisothermal Cahn–Hilliard–Navier–Stokes systems couple velocity to a conserved phase field $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$5, chemical potential $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$6, and inverse temperature $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$7. Their momentum equation includes viscous, pressure, and Korteweg stresses:

$$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$8

The continuous model conserves phase mass and total energy while producing entropy. A stabilized equal-order $$\frac{D\mathbf V}{Dt} = \frac{\partial\mathbf V}{\partial t} +\mathbf V\cdot\nabla\mathbf V,$$9 finite-element scheme uses Brezzi–Pitkäranta pressure stabilization and grad-div stabilization. Matching stabilization terms are inserted into the internal-energy equation so that artificial kinetic contributions cancel in the total-energy balance and reappear as nonnegative numerical entropy production (Brunk et al., 2024).

6. Boundary problems, steady flows, and numerical methods

For steady incompressible flow,

$$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$0

Liouville-type results classify bounded smooth solutions under geometric or boundary restrictions. In $$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$1, bounded smooth helically symmetric solutions are constant vertical flows:

$$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$2

In the slab $$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$3 with Navier boundary conditions,

$$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$4

bounded solutions vanish under axisymmetry of the radial or swirl component, or under suitable radial-flux decay. For sufficiently small $$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$5 norm, solutions reduce to horizontal Poiseuille-type profiles. The proofs use localized Dirichlet integrals, Bogovskii corrections for the pressure, and Saint–Venant growth inequalities (Han et al., 2023).

Time-discrete variational schemes approximate material differentiation by

$$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$6

The new velocity minimizes a transported $$\mathcal N\mathbf V = \frac{D\mathbf V}{Dt}-\mathbf F_H=0,$$7 distance together with Dirichlet energy. The method yields a discrete energy estimate, but the cited construction does not fully establish convergence to the complete Navier–Stokes weak formulation because the displayed limiting equation omits the nonlinear convection term and several compactness and initial-data arguments are incomplete (Horihata, 2014).

Other numerical approaches use space-time least squares, damped Newton iteration, finite elements, and implicit time discretization. In two dimensions, the least-squares functional is globally convergent analytically from arbitrary admissible initial guesses and becomes quadratically convergent after a finite damped phase. In three dimensions, the corresponding theory is local in time for regular solutions (Lemoine et al., 2019).

The Navier–Stokes system also appears in open thermodynamic domains. For compressible Navier–Stokes–Fourier flow with inflow and outflow, density is prescribed only on inflow, energy flux is prescribed on inflow, and heat flux is insulated on walls and outflow. Because density lacks a classical outflow trace and the boundary heat flux is not stable in the weak topology, the weak theory uses entropy and ballistic-energy inequalities rather than the standard total-energy identity. A relative-energy inequality yields weak–strong uniqueness (Feireisl et al., 2020, Chaudhuri et al., 2021).

Across these formulations, several distinctions are essential. A finite-dimensional characteristic or particle representation does not imply finite-dimensionality of the Eulerian field problem. A special explicit solution does not establish general global regularity. A forced singular solution smooth away from a curve is not a singular solution of the unforced equations. A numerical stabilization preserving a discrete energy law is not by itself a convergence theorem. Finally, weak existence, uniqueness, regularity, and long-time behavior are separate properties: existing results establish different subsets of them under different dimensional, constitutive, geometric, stochastic, and boundary assumptions.