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navier stokes equation

1. Why this matters (Millennium Problem)

Fluids are everywhere—stars, oceans, air, plasmas.
Yet the 3D incompressible Navier–Stokes equations still lack a proof of global smooth existence & uniqueness for smooth data (Clay Millennium Problem).
Understanding regularity also underpins turbulence, arguably the last great classical physics problem.

2. Continuum Fields and Conservation Laws

Let ρ(x,t)\rho(\boldsymbol x,t) be mass density, v(x,t)\boldsymbol v(\boldsymbol x,t) velocity, p(x,t)p(\boldsymbol x,t) pressure, and σ\boldsymbol{\sigma} the Cauchy stress. Body force per mass f\boldsymbol f (e.g. gravity).

2.1 Mass conservation

∂tρ+∇ ⁣⋅(ρv)=0.\partial_t \rho + \nabla\!\cdot(\rho \boldsymbol v) = 0.

2.2 Momentum conservation (Cauchy)

ρ(∂tv+(v ⁣⋅ ⁣∇)v)=∇ ⁣⋅σ+ρf.\rho\left(\partial_t \boldsymbol v + (\boldsymbol v\!\cdot\!\nabla)\boldsymbol v\right) = \nabla\!\cdot \boldsymbol{\sigma} + \rho \boldsymbol f .

Decompose stress into isotropic pressure and deviatoric part:

σ=−p I+τ.\boldsymbol{\sigma} = -p\,\boldsymbol I + \boldsymbol{\tau}.

3. Constitutive Law (Newtonian Fluid)

For an isotropic Newtonian fluid,

τ=2μ S+λ (∇ ⁣⋅ ⁣v) I,S≡12(∇v+∇v ⁣⊤),\boldsymbol{\tau} = 2\mu\,\boldsymbol S + \lambda\,(\nabla\!\cdot\!\boldsymbol v)\,\boldsymbol I, \qquad \boldsymbol S \equiv \tfrac12\left(\nabla \boldsymbol v + \nabla \boldsymbol v^{\!\top}\right),

with dynamic viscosity μ\mu, bulk viscosity λ\lambda.

Plugging into momentum balance gives the compressible Navier–Stokes:

ρ(∂tv+v ⁣⋅ ⁣∇v)=−∇p+μ∇2v+(μ+λ) ∇(∇ ⁣⋅ ⁣v)+ρf.\rho\left(\partial_t \boldsymbol v + \boldsymbol v\!\cdot\!\nabla \boldsymbol v\right) = -\nabla p + \mu \nabla^2 \boldsymbol v + (\mu+\lambda)\,\nabla(\nabla\!\cdot\!\boldsymbol v) + \rho \boldsymbol f .

Define kinematic viscosity ν≡μ/ρ\nu \equiv \mu/\rho when ρ\rho is constant.

4. Incompressible Form

For constant density ρ=ρ0\rho=\rho_0 and

∇ ⁣⋅ ⁣v=0,\nabla\!\cdot\!\boldsymbol v = 0,

we obtain

∂tv+(v ⁣⋅ ⁣∇)v=−1ρ0 ∇p+ν∇2v+f,∇ ⁣⋅ ⁣v=0.\partial_t \boldsymbol v + (\boldsymbol v\!\cdot\!\nabla)\boldsymbol v = -\tfrac{1}{\rho_0}\,\nabla p + \nu \nabla^2 \boldsymbol v + \boldsymbol f, \qquad \nabla\!\cdot\!\boldsymbol v = 0.

Taking divergence of the momentum equation yields a pressure Poisson equation (with suitable BCs):

∇2p=−ρ0 ∇ ⁣⋅ ⁣[(v ⁣⋅ ⁣∇)v]+ρ0 ∇ ⁣⋅ ⁣f.\nabla^2 p = -\rho_0\,\nabla\!\cdot\!\big[(\boldsymbol v\!\cdot\!\nabla)\boldsymbol v\big] + \rho_0\,\nabla\!\cdot\!\boldsymbol f .

5. Vorticity and Helicity

Define vorticity ω≡∇×v\boldsymbol{\omega} \equiv \nabla \times \boldsymbol v.
For incompressible flow (∇ ⁣⋅ ⁣v=0\nabla\!\cdot\!\boldsymbol v=0):

∂tω=∇×(v×ω)+ν∇2ω+∇×f.\partial_t \boldsymbol{\omega} = \nabla\times(\boldsymbol v \times \boldsymbol{\omega}) + \nu \nabla^2 \boldsymbol{\omega} + \nabla \times \boldsymbol f .

In components:

∂tωi+vj∂jωi=ωj∂jvi+ν∂jjωi+(∇×f)i.\partial_t \omega_i + v_j\partial_j \omega_i = \omega_j \partial_j v_i + \nu \partial_{jj} \omega_i + (\nabla\times\boldsymbol f)_i .

The term ωj∂jvi\omega_j \partial_j v_i is vortex stretching (3D only).
Helicity H≡∫v ⁣⋅ ⁣ω dVH\equiv\int \boldsymbol v\!\cdot\!\boldsymbol{\omega}\,dV diagnoses topology of vortex lines.

6. Energy Balance (Incompressible)

Multiply momentum by v\boldsymbol v and integrate over a domain Ω\Omega with no-slip at ∂Ω\partial\Omega:

12ddt∫Ω∣v∣2 dV+ν∫Ω∣∇v∣2 dV=∫Ωf ⁣⋅ ⁣v dV.\tfrac12 \tfrac{d}{dt}\int_\Omega |\boldsymbol v|^2\,dV + \nu \int_\Omega |\nabla \boldsymbol v|^2\,dV = \int_\Omega \boldsymbol f\!\cdot\!\boldsymbol v\,dV .

For f=0\boldsymbol f=\boldsymbol 0, kinetic energy decays monotonically; viscosity dissipates as ν∥∇v∥22\nu\|\nabla \boldsymbol v\|_2^2.

7. Non-dimensionalization and Control Parameters

Choose characteristic scales LL (length), UU (speed), define x′=x/L\boldsymbol x'=\boldsymbol x/L, t′=tU/Lt' = tU/L, v′=v/U\boldsymbol v'=\boldsymbol v/U, p′=p/(ρU2)p'=p/(\rho U^2).

The incompressible equations become

∂t′v′+(v′ ⁣⋅ ⁣∇′)v′=−∇′p′+1Re∇′2v′+f′,∇′ ⁣⋅ ⁣v′=0,\partial_{t'} \boldsymbol v' + (\boldsymbol v'\!\cdot\!\nabla')\boldsymbol v' = -\nabla' p' + \tfrac{1}{\mathrm{Re}}\nabla'^2 \boldsymbol v' + \boldsymbol f', \quad \nabla'\!\cdot\!\boldsymbol v' = 0,

with the Reynolds number

Re=ULν.\mathrm{Re} = \frac{U L}{\nu}.

Other dimensionless groups (as needed):

  • Mach Ma=U/cs\mathrm{Ma}=U/c_s (compressibility),
  • Froude Fr=U/gL\mathrm{Fr}=U/\sqrt{gL} (gravity),
  • Prandtl Pr=ν/κ\mathrm{Pr}=\nu/\kappa (momentum vs thermal diffusion).

8. Compressible Navier–Stokes–Fourier (Sketch)

Include internal energy ee or temperature TT, with Fourier heat flux q=−κ∇T\boldsymbol q=-\kappa \nabla T:

∂t(ρe)+∇ ⁣⋅(ρe v)=−p ∇ ⁣⋅ ⁣v+τ:∇v−∇ ⁣⋅ ⁣q.\partial_t (\rho e) + \nabla\!\cdot(\rho e\,\boldsymbol v) = -p\,\nabla\!\cdot\!\boldsymbol v + \boldsymbol{\tau}:\nabla \boldsymbol v - \nabla\!\cdot\!\boldsymbol q .

Equation of state (e.g., ideal gas) closes the system: p=(γ−1)ρep=(\gamma-1)\rho e.

9. Mathematical Theory (Existence/Uniqueness)

9.1 Weak solutions (Leray–Hopf)

For 3D incompressible NS with L2L^2 data, global-in-time weak solutions exist that satisfy the energy inequality.
Uniqueness of such weak solutions is open.

9.2 Regularity criteria (3D)

  • Beale–Kato–Majda: blow-up at TT implies
∫0T∥ω(⋅,t)∥L∞ dt=∞.\int_0^T \|\boldsymbol{\omega}(\cdot,t)\|_{L^\infty}\,dt = \infty .
  • Prodi–Serrin conditions: if v∈Lp(0,T;Lq)\boldsymbol v \in L^p(0,T;L^q) with
2p+3q≤1,q>3,\frac{2}{p} + \frac{3}{q} \le 1,\quad q>3,

then the solution is smooth on (0,T](0,T].

  • Caffarelli–Kohn–Nirenberg: suitable weak solutions have a singular set in spacetime of parabolic Hausdorff dimension ≤1\le 1 (partial regularity).

9.3 Global regularity in 2D

For smooth data, global existence and uniqueness hold in 2D.
Enstrophy ∫∣ω∣2dV\int |\boldsymbol{\omega}|^2 dV remains bounded; no vortex stretching.

9.4 Compressible flows

Global finite-energy weak solutions exist (for γ\gamma-law gases under conditions), but regularity/uniqueness are largely open.

Millennium Problem: Prove or give a counterexample that in 3D, smooth, divergence-free initial data yield a unique smooth global solution (or exhibit finite-time singularity).

10. Turbulence and Spectra

In high-Re\mathrm{Re} homogeneous, isotropic turbulence (K41), the inertial-range energy spectrum

E(k)=CK ε2/3k−5/3,E(k) = C_K\,\varepsilon^{2/3} k^{-5/3},

with mean dissipation rate ε\varepsilon and Kolmogorov constant CK ⁣∼ ⁣O(1)C_K\!\sim\! \mathcal O(1).

Kolmogorov microscale:

η∼(ν3ε)1/4.\eta \sim \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}.

DNS cost scales steeply (grid points ∼Re9/4\sim \mathrm{Re}^{9/4} for 3D), motivating LES and RANS closures; the closure problem (modeling subgrid stresses) remains central.

11. Boundary Conditions & Geometry

  • No-slip wall: v∣∂Ω=0\boldsymbol v|_{\partial\Omega}=\boldsymbol 0
  • Free-slip: n ⁣⋅ ⁣v=0\boldsymbol n\!\cdot\!\boldsymbol v=0, n ⁣⋅ ⁣τ ⁣⋅ ⁣t=0\boldsymbol n\!\cdot\!\boldsymbol{\tau}\!\cdot\!\boldsymbol t=0
  • Periodic box (torus) for theory/DNS
  • Inflow/Outflow with appropriate stress/pressure specs

Choice affects well-posedness, energy budget, and numerical stability.

12. Numerics (one-page field guide)

  • Projection methods: advance v∗\boldsymbol v^\ast, solve Poisson for pp, project to ∇ ⁣⋅ ⁣vn+1=0\nabla\!\cdot\!\boldsymbol v^{n+1}=0.
  • Energy-stable schemes: skew-symmetric convective form preserves discrete energy.
  • CFL constraint: Δt≲min⁡{Δx/U,Δx2/ν}\Delta t \lesssim \min\{\Delta x/U, \Delta x^2/\nu\}.
  • Divergence-free bases: spectral/finite-element methods with inf–sup stable pairs.

13. Connections and Extensions

  • Euler limit: ν→0\nu\to 0 → incompressible Euler; possible finite-time singularities are a parallel grand challenge.
  • Relativistic hydrodynamics: naive Navier–Stokes is acausal; Israel–Stewart (second-order) fixes causality.
  • Quantum fluids: superfluid N-S analogues with two-fluid models and quantized vortices.

14. Status

  • 2D: theory is mature; global smoothness known.
  • 3D: existence of global smooth solutions or finite-time blow-up is open.
  • Turbulence phenomenology (K41) is broadly successful but intermittency and closures remain active research.

TL;DR: We can simulate and measure astonishingly well, but the deep math—regularity and turbulence—still hides the final boss.