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The biharmonic equation is ${\nabla }^{4}f=0$, where the biharmonic operator is the square of the Laplacian. In planar Stokes flow, taking the curl of the momentum equation and using a Cartesian streamfunction produces this equation. Its decaying spatial Fourier series modes have form $(A+By)e^{-|k|y}{e}^{ikx}$.

The biharmonic operator applies the Laplacian twice. Under clamped boundary conditions, two integrations by parts give the positive quadratic form

Although $1/r$ is harmonic away from the origin in three dimensions, it is not a regular harmonic function on a ball containing the origin. Its gradient is $-\hat{r}/r^2$, so the displayed improper volume integral follows by canceling the $r^2$ in the spherical volume element. On the annulus $\epsilon <r<a$, Green's first identity gives outer boundary contribution $-4\pi$ and inner contribution $4\pi \epsilon /a$, recovering the same limit. Its global Laplacian contains a Dirac delta distribution, so the regular-harmonic gradient-pairing theorem is inapplicable.

Writing $D_r=\mathbf{x}\cdot \nabla =r{\partial }_r$, the spherical-coordinate Laplacian gives $r^2\Delta =D_r^2+D_r+{\Delta }_{S^2}$. Thus the positive angular Laplace-Beltrami operator has the displayed Cartesian expression and eigenvalues $l(l+1)$ on spherical harmonics. Omitting $D_r$ is incorrect when the square means composition: on $x_1$, the truncated expression gives $x_1$, whereas the degree-one angular eigenvalue is two.

// Target: fluid-mechanics.bigb

This is the closure of the compactly supported test functions $C_c^{\infty }(U)$ in the $H^2(U)$ norm. On a smooth bounded domain it consists precisely of the $H^2$ functions with zero value and zero normal derivative traces. Tangential first derivatives then vanish as well. The clamped Hessian identity makes the Laplacian norm equivalent to the full $H^2$ norm on this space, enabling a Lax-Milgram theorem treatment of the clamped biharmonic problem.

A symmetric uniformly elliptic divergence-form energy with zeroth-order coefficient at least $\mu >0$ controls the full $H^1$ norm, including constants. Its quadratic energy minus a bounded load has a unique minimizer by the direct method, and vanishing first variation gives the natural Neumann weak equation. Unlike a pure Neumann Laplacian, this problem needs no zero-mean compatibility condition. The exact energy difference from the weak solution is half the bilinear energy of the difference.

If $H,G$ are holomorphic functions, both real and imaginary parts of $\bar{z}H(z)+G(z)$ solve the biharmonic equation. In Wirtinger derivatives, $\Delta =4{\partial }_z{\partial }_{\bar{z}}$, so $\Delta (\bar{z}H+G)=4H^{\prime }$ is harmonic and its next Laplacian is zero. This construction gives local representations of planar solutions of the biharmonic equation; it is useful for the biharmonic stream function for planar Stokes flow.

A vector field in Euclidean space is componentwise harmonic when each Cartesian component is a harmonic function, equivalently its componentwise Laplacian vanishes. Constant and linear vector fields are examples, as are constant linear combinations of derivatives of $1/r$ away from the origin in three dimensions. Such fields supply the vector potentials in the Papkovich–Neuber representation. Componentwise harmonicity does not by itself require zero divergence or zero curl.

For a conformal map extending to the relevant boundary sets, compose the target harmonic measure with the map. The planar Laplacian transforms by the positive factor $|f^{\prime }{|}^2$, so the composition is a harmonic function with the correct boundary values. Uniqueness of the Dirichlet problem gives the displayed identity on regular domains, including the smooth-circle domains in the harmonic measure of an annulus.

For $g^{\prime }(0)\ne 0$, apply Jensen's formula with the smooth curvature term to the hyperbolic derivative density. Its distributional Laplacian is $H_g^2+2\pi \sum m_a{\delta }_a$, giving the displayed identity with ramification points defined by $g^{\prime }(a)=0$. If $g^{\prime }(z)=c{z}^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log {\lim }_{z\to 0}{H}_g(z)/|z{|}^m$. Without this regularization the basepoint-critical formula is undefined.

This collects the gradient, divergence, curl and Laplacian formulas in cylindrical and spherical coordinates. Coordinate unit vectors vary with position, so taking their derivatives is essential when translating vector calculus operators from Cartesian coordinates. Curl in spherical coordinates is one of these formulas.

A differential operator maps a function to an expression depending on finitely many of its derivatives. A linear differential operator has the form $P={\sum }_{|\alpha |\le m}{a}_{\alpha }(x){\partial }^{\alpha }$; its order is the largest derivative order with nonzero coefficient. The Laplacian is a second-order example, and boundary conditions specify the domain when defining an operator on a function space.

Diffusion is net transport associated with random microscopic motion. A localized patch of a dilute solute can spread through a stationary solvent: individual molecules move in both directions, but more cross from the crowded side of an imaginary surface than from the less crowded side. This imbalance gives a net diffusive flux down the concentration gradient; the molecules do not need to sense which direction has lower concentration. At uniform concentration, microscopic exchanges continue even though the net diffusive flux vanishes. ^[ref]

The distinction from advection is easiest to see in a dye patch. Uniform bulk motion carries its centre downstream; diffusion broadens it relative to that centre. Both can happen together. A moving or spreading patch by itself does not identify the mechanism: a nonuniform velocity field can also stretch a patch. ^[ref]

For a dilute nonreacting solute in a uniform medium at fixed temperature, Fick's first law models the relative diffusive flux as $\mathbf{j}=-D\nabla C$, with scalar diffusion coefficient $D>0$ of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Here $C$ is the amount of substance per unit volume, so $\mathbf{j}$ has units $\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. With carrier velocity field $\mathbf{u}$, the total flux is $C\mathbf{u}+\mathbf{j}$. The continuity equation, with no sources or chemical reactions, givesFor constant $D$ and incompressible flow, this reduces to $C_t+\mathbf{u}\cdot \nabla C=D{\nabla }^{2}C$, where ${\nabla }^2$ is the Laplacian. In a stationary carrier it is the diffusion equation. These assumptions matter: variable $D$ must remain inside the divergence, and changing sources require extra terms. ^[ref]

The useful surprise is the common mathematics: thermal conduction smooths temperature through the analogous heat equation, while molecular diffusion smooths composition. This analogy already motivated the original formulation of Fick's laws. ^[ref] However, the thermal diffusivity and the solutal diffusivity need not be equal. A diffusive time scale $t_D\sim L^2/D$ means that crossing ten times the distance requires roughly one hundred times the time. This is why molecular diffusion is especially effective across thin layers, and why chaotic advection can assist mixing by making those layers thinner. Physical diffusion is a transport mechanism, distinct from the broader probabilistic class of Markov diffusion.

A finite-dimensional approximation of the Laplacian. On a periodic one-dimensional grid the centered operator has Fourier eigenvalue $-4{\sin }^2(\theta /2)/h^2$ and a constant nullspace.

On $0<x<\ell$, homogeneous Neumann boundary conditions give spatial Fourier modes $\cos (m\pi x/\ell )$, $m=0,1,\dots$, with squared wavenumbers $q_m$. Differentiating twice proves that their Laplacian eigenvalues are $-q_m$, and their endpoint derivatives vanish. A dispersion relation $\sigma (q)$ therefore predicts the fastest permitted growing mode by maximizing $\mathrm{Re}\sigma (q_m)$ over integers $m$, rather than over all real wavenumbers. A continuous unstable band is insufficient if it contains no $q_m$.

For a semiclassical wave packet in an isolated energy band, $v_i={\hslash }^{-1}{\partial }_{k_i}\epsilon$ and $\hslash {\dot{k}}_j=F_j$ imply ${\dot{v}}_i={\sum }_j(m^{\ast -1}{)}_{ij}{F}_j$. The inverse effective-mass tensor is the Hessian matrix of the dispersion divided by ${\hslash }^2$. Along a unit vector $\mathbf{e}$, $1/m_{\mathbf{e}}^{\ast }={\sum }_{ij}{e}_i(m^{\ast -1}{)}_{ij}{e}_j$. The Laplacian is its trace: ${\hslash }^{-2}{\nabla }_k^2\epsilon =\mathrm{tr}(m^{\ast -1})$. Thus in $d$ isotropic dimensions the scalar mass is $d{\hslash }^2/({\nabla }_k^2\epsilon )$, rather than ${\hslash }^2/({\nabla }_k^2\epsilon )$.

In source-free vacuum the electric field and magnetic field obeyA brief derivation takes the curl of Faraday's law and substitutes the Ampère-Maxwell equation. The vector calculus identity $\nabla \times (\nabla \times \mathbf{E})=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$, together with source-free Gauss's law, leaves ${\nabla }^2\mathbf{E}={\mu }_0{ϵ}_0{\partial }_t^2\mathbf{E}$. The same argument gives the magnetic field equation. The operator ${\nabla }^2$ is the Laplacian, applied componentwise.

Elliptic regularity says that solutions of an elliptic equation gain derivatives wherever the coefficients and forcing are regular. For the Laplacian, local estimates typically control two derivatives of $u$ by $\Delta u$ together with a lower-order norm of $u$.

For a straight pipe with cross-section $0<x<a$, $|y|<x/\sqrt{3}$, steady unidirectional incompressible flow reduces the Navier-Stokes equations to $\mu {\nabla }_{\perp }^{2}w=p_z$, with constant axial pressure gradient. The displayed cubic polynomial vanishes on every wall, and its transverse Laplacian is $p_z/\mu$. The axial drag per unit length on each of the three equal walls is $-p_z{a}^2/(3\sqrt{3})$. This follows by integrating the viscous traction on one wall; summing the three forces equals the pressure gradient times the triangular area.

For positive background velocity $U=U_0{e}^{Mz}$ and buoyancy frequency $N=N_0{e}^{Mz}$, a stationary nonzero-horizontal-wavenumber Taylor–Goldstein equation reduces to the displayed constant-coefficient equation. Its structures are oscillatory, affine or exponential according to the sign of $N_0^2/U_0^2-M^2-k^2$. The disturbance Laplacian is a constant times its streamfunction, but its buoyancy anomaly is $N_0^2{e}^{Mz}\varphi /U_0$. Therefore vorticity self-advection vanishes while buoyancy self-advection produces a nonzero second harmonic for $M\ne 0$. These nontrivial real linear modes are not exact finite-amplitude solutions without additional fields. For $M=0$, both polarizations are constant and these nonlinear self-interactions vanish.