Diffusion equation

A diffusion equation describes smoothing caused by a flux down spatial gradients.

Infinite propagation speed means that a disturbance initially confined to a bounded region can affect arbitrarily distant points after any positive time. The strictly positive heat kernel gives this behavior for a nontrivial nonnegative diffusion solution. It contrasts with the finite propagation speed of the wave equation; a very small distant response is still mathematically nonzero.

For $u_t={\partial }_x[a(x/ϵ)u_x]$ with positive periodic $a$, the one-dimensional homogenized diffusivity is the harmonic mean $a_{\mathrm{eff}}=⟨a^{-1}{⟩}^{-1}$. Microscopic flux continuity produces this harmonic rather than arithmetic averaging.

The porous medium equation $u_t=\Delta (u^m)$ with $m>1$ is a nonlinear partial differential equation whose diffusivity vanishes at $u=0$. Unlike the heat equation, it has compactly supported solutions with finite propagation speed.

In a porous medium equation with exponent $m>1$, the effective diffusion coefficient vanishes at zero density. A nonnegative compactly supported source profile can therefore retain a finite front for every finite positive time. For example, the radial cubic-diffusion source profile has support radius proportional to $t^{1/6}$. This notion concerns a moving diffusion front; it does not require the constant characteristic cone used for a wave equation.

Consider a nonlinear diffusion equation with uniform supply, $\varphi h_t=D_m(h^m{h}_x{)}_x+R$, on $x>0$, with an absorbing boundary $h(0,t)=0$ and initially $h=0$. Far from the boundary, $h=Rt/\varphi$. Balancing the time derivative, supply, and diffusion givesThe similarity solution satisfiesThe outward boundary volume flux per unit width is $Q=R\ell c_m$, where $c_m={\lim }_{\eta \downarrow 0}{F}^m{F}^{\prime }$. Integrating the equation givesThus the growing region of depleted storage fixes the discharge prefactor, and $Q\propto t^{(m+1)/2}$.

For $\varphi h_t=D_m(h^m{h}_x{)}_x$ on $0<x<L$, with $h(0,t)=0$ and zero right-hand volume flux, a separation of variables giveswhere the positive profile satisfiesIts boundary volume flux per unit width isThese profiles are exact separated solutions and describe the leading long-time discharge for a broad class of positive initial data. The virtual origin $t_0$ depends on the initial profile; it does not make the separated solution an exact representation of every initial condition.

The Barenblatt solution is a mass-preserving similarity solution of the porous medium equation. For $u_t=\beta (u{u}_x{)}_x$ in one dimension it has the form

For the planar porous medium equation $n_t=[D_0/(3n_0^2)]\Delta (n^3)$, the mass-$Q$ point-source similarity solution is $n=n_0{\lambda }^{-2}\sqrt{[1-r^2/(r_0^2{\lambda }^2){]}_{+}}$, with $\lambda =(6D_{0}t/r_0^2{)}^{1/6}$ and $r_0^2=3Q/(2\pi n_0)$. Its front grows as $t^{1/6}$ and its central density falls as $t^{-1/3}$. The mass conservation integral uses the planar area element $2\pi rdr$. Although the density slope diverges at the front, its diffusion flux vanishes there, giving a compactly supported weak solution with initial measure $Q{\delta }^{(2)}$.

For $n,A>0$, the one-dimensional porous medium equation $h_t=A(h^n{h}_x{)}_x$ has a volume-preserving similarity solution $h=[c_n(R^2-x^2)/(At){]}_{+}^{1/n}$, where $c_n=n/[2(n+2)]$ and $R\propto t^{1/(n+2)}$. On a reflecting half-line with conserved area $M$, $R=[M(At/c_n{)}^{1/n}/I_n{]}^{n/(n+2)}$, where $I_n=\frac{1}{2}B(\frac{1}{2},1+1/n)$. For a whole symmetric release use half its total area as $M$. The Beta function fixes normalization, while finite-front zero volume flux fixes the edge condition.

The source-type similarity solution of $h_t=D(h^2{h}_x{)}_x$ with whole-line conserved area $\mathcal{V}$ is $h=H(t)[1-x^2/L(t{)}^2{]}_{+}^{1/2}$, where $L=2(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{1/4}$ and $H=(\mathcal{V}/\pi {)}^{1/2}(Dt{)}^{-1/4}$. Its volume flux vanishes at the finite front, while the velocity tends to $\dot{L}=L/(4t)$. A reflecting half-line uses twice its prescribed area in these formulas.

For the radially symmetric equationin two dimensions, the mass-$N$ similarity solution has the compactly supported formwhere mass normalization gives $x_0=(16/\pi {)}^{1/4}$. Its support radius grows as $t^{1/4}$ and its occupied area grows as $t^{1/2}$.

Fick's first law says that diffusive flux points down the concentration gradient:Combining it with local conservation gives
$C_t=\nabla \cdot (D(C)\nabla C)$.

The heat equationmodels diffusion with constant diffusivity $D$.

Classical heat equation solutions with identical continuous initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\epsilon {\Phi }_b$, where ${\Phi }_b=(1-4bt{)}^{-n/2}\exp (b|x{|}^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The heat equation maximum principle bounds both signs of the difference by $\epsilon {\Phi }_b$ inside. Increasing the cylinder radius and then sending $\epsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.

A Fourier mode has multiplier $e^{D{k}^{2}t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-D{n}^2{t}_0/2}\sin (nx)$ tend to zero in $L^2$ while their values at $t_0$ grow without bound. This violates continuous dependence on initial data and gives an ill-posed problem in ordinary unweighted spaces. An analytic-data restriction or physical short-scale regularization changes that conclusion.

On a closed manifold with a Riemannian metric, the minimum of a smooth heat solution cannot fall below its initial minimum. At a spatial minimum, $\Delta u\le 0$ for the positive Laplace-Beltrami operator. Adding a small increasing function of time and considering the first crossing gives the assertion. Applying the result to nonnegative initial functions shows that the Riemannian heat kernel is nonnegative.

On a closed manifold with a Riemannian metric, a smooth solution of ${\partial }_{t}u+\Delta u=0$ for the positive Laplace-Beltrami operator satisfies the displayed identity by integration by parts. Zero initial data force the solution to vanish, proving uniqueness and the semigroup property for heat evolution.

If $K$ is the zero-Dirichlet heat kernel on an interval for $w_t=w_{xx}-cw$, integration by parts givesThe opposite endpoint signs are essential. Nonzero boundary values are limits of the full formula; evaluating each homogeneous sine mode at the boundary prematurely loses the forcing.

Convolution of this kernel with Dirichlet boundary data supplies the boundary forcing for $u_t=u_{xx}+\alpha u_x$ on $x>0$. It is $-(2{\partial }_x+\alpha )H(x+\alpha t,t)$. For $\alpha \ge 0$, its total time mass is $e^{-\alpha x}$, tending to one as $x\downarrow 0$, and its mass away from zero time tends to zero. Thus the boundary value is recovered through an approximate identity, not by pointwise substitution $x=0$ in the kernel.

For a real smooth solution $w_t=\Delta w$ on a bounded regular domain with $w_t=0$ on its boundary, differentiation and Green's first identity yieldThe boundary term is zero because the boundary values are fixed in time, not because the normal derivative vanishes. Equality at a particular time is equivalent to $\Delta w=0$ throughout the domain at that time.

For a spherically symmetric field $u(r,t)$ in three dimensions,Thus the heat equation becomes $u_t=D{r}^{-1}(ru{)}_{rr}$. Introducing $w=ru$ reduces its spatial part to the one-dimensional second derivative $w_{rr}$, while regularity at the centre requires $w(0,t)=0$.

For $T_t=\kappa T_{xx}$ on $x>0$, with $T(x,0)=0$ and $T(0,t)=\sin \omega t$,Consequently the transform of $I(t)={\int }_0^{\infty }T(x,t)dx$ is

The nonlinear equationis the potential form of the viscous Burgers equation. The substitution $w=e^u$ converts it to the heat equation $w_t=w_{xx}$.

For $u_t=\kappa u_{xx}$ on the real line,has total mass one and converges to the delta distribution as $t$ decreases to zero. The solution is $u(t)=K_t\ast u_0$.

The real Gaussian function heat kernel $(4\pi \epsilon z{)}^{-1/2}\exp [-(\varphi -\theta {)}^2/(4\epsilon z)]$ is integrable only when $\epsilon z>0$. For forward time $z>0$, this requires positive diffusivity. A Cole-Hopf transformation is an algebraic substitution for any nonzero coefficient, but that does not make the real forward convolution valid for negative diffusivity. Constant initial heat data already produce a divergent integral with the wrong sign.

For the heat equation on ${\mathbb{R}}^n$ with diffusion coefficient one, the heat kernel is $K_t(x)=(4\pi t{)}^{-n/2}{e}^{-|x{|}^2/(4t)}$. Its convolution with bounded initial data is smooth for positive time, solves the equation by differentiation under the integral, and returns continuous initial data locally uniformly through the approximate identity property. It is Gaussian filtering with per-coordinate variance $2t$.

For the positive Laplace-Beltrami operator, the kernel solving $({\partial }_t+{\Delta }_x)H=0$ with initial Dirac delta distribution on the diagonal represents the heat semigroup. On a closed manifold with a Riemannian metric it is smooth for $t>0$, symmetric, nonnegative, preserves constants, and obeys the semigroup property.

For a finite free action by Riemannian isometries, the Riemannian heat kernel on the quotient is the image sum shown above. Integrating over a fundamental domain combines all images into the integral on the covering manifold, proving the initial condition and showing that there is no averaging factor in the kernel. The heat trace does have an averaging factor $1/|U|$, because the integral of a quotient function over the cover is $|U|$ times its quotient integral. Mere freeness of a general nondiscrete group action is not enough for this covering formula.

On a closed manifold with a Riemannian metric, an orthonormal eigenbasis for the positive Laplace-Beltrami operator gives this formula, with eigenvalues counted with multiplicity. Polynomial elliptic bounds and exponential time decay give convergence of every derivative away from time zero. The conjugate is necessary for a complex basis.

An approximate Riemannian heat kernel with the correct delta initial limit and a residual extending with enough regularity to time zero. A cutoff Gaussian times transport coefficients supplies a local construction. Increasing the expansion order, or summing the full asymptotic series smoothly, makes the residual regular enough for a Volterra parametrix correction.

If $R=({\partial }_t+{\Delta }_x)P$ is smooth and bounded up to time zero on a closed manifold, the Volterra convolution of kernels solves $Q+R+R\ast Q=0$ by $Q={\sum }_{k\ge 1}(-1{)}^k{R}^{\ast k}$. Bounds by $C^k\mathrm{vol}(M{)}^{k-1}{t}^{k-1}/(k-1)!$ prove convergence. Then $H=P+P\ast Q$ is the exact Riemannian heat kernel and has the same initial delta limit.

The normalized Gaussian mass of an interval isIt is the convolution of an interval indicator with the heat kernel, and has an error function representation. As the variance tends to zero it converges to the interval indicator away from its endpoints, with value one half at an endpoint. This is a concrete approximate identity.

For $x>0$ and the heat equation on a half-line, the zero-initial-data solution with prescribed Dirichlet boundary condition $g$ is ${\int }_0^{t}P(x,t-s)g(s)ds$, whereIts mass concentrates at $t=0$ as $x\downarrow 0$, recovering the boundary trace.

For the heat equation with zero Dirichlet boundary conditions on $(0,L)$,It is also obtained by the method of images, taking the difference between Gaussian sources at $y+2jL$ and $-y+2jL$. The eigenfunction expansion is useful at long times; the image expansion is useful at short times.

For a particle with mass $m$ on $(0,L)$, put $K_0(x;\beta )=\sqrt{m/(2\pi {\hslash }^2\beta )}{e}^{-m{x}^2/(2{\hslash }^2\beta )}$. The method of images gives $K_D(q_f,q_i;\beta )={\sum }_r[K_0(q_f-q_i+2rL;\beta )-K_0(q_f+q_i+2rL;\beta )]$. The diagonal integral isThe reflected intervals tile the real line, giving the subtraction $1/2$. The Poisson summation formula converts this to ${\sum }_{n\ge 1}{e}^{-\beta {\hslash }^2{\pi }^2{n}^2/(2m{L}^2)}$, proving equality between the image and energy eigenstate calculations.

For standard Brownian motion in ${\mathbb{R}}^d$, $p_t(y-x)$ is the transition density from $x$ to $y$. It is the heat kernel for generator $\frac{1}{2}\Delta$. For every unit vector $e$, $\int |{\partial }_e{p}_t(x)|dx=\sqrt{2/\pi }/\sqrt{t}$, which bounds the change in this density under a spatial translation and proves the Gaussian heat-kernel proof of the harmonic Liouville theorem.

For $L>0$ and $t>0$, the Sturm-Liouville eigenfunction expansion for $u_t=u_{xx}$ on $(0,L)$ with homogeneous Neumann boundary conditions givesThe constant eigenfunction mode preserves the spatial integral. This is the $\beta =0$ limit of the weighted Neumann heat kernel with constant drift, and its boundary inputs have the signs in the Neumann boundary-forcing formula.

The heat semigroup acts by convolution with the heat kernel: $P_{t}f=K_t\ast f$. The identity $K_s\ast K_t=K_{s+t}$ gives $P_s{P}_t=P_{s+t}$, and its infinitesimal generator is a constant multiple of the Laplacian.

For a time-independent strictly positive Dirichlet realization of an elliptic operator and $\psi \in L^2(U)$, the eigenfunction expansionsolves $u_t+Lu=0$ with homogeneous Dirichlet boundary conditions. For $t\ge \epsilon >0$, every power ${\lambda }_m^N$ times the exponential is bounded, so the series and all its time derivatives converge in all the Sobolev domains of powers of an elliptic Dirichlet operator. Elliptic regularity and the Sobolev embedding theorem give smoothness up to the spatial boundary for positive time. Parseval identity and the dominated convergence theorem give $u(t)\to \psi$ in $L^2$ as $t\downarrow 0$.

A heat kernel expansion is the short-time asymptotic expansion of the trace or diagonal kernel of an elliptic operator. Its local coefficients are polynomials in background curvatures and field strengths; in quantum field theory they extract ultraviolet divergences and anomalies.

The centered normal densitiesform an approximate identity: for every $f\in L^1(\mathbb{R})$, $f\ast g_t\to f$ at almost every point where the Lebesgue differentiation theorem applies. Their Fourier transforms are $e^{-t{\xi }^2/2}$ in the angular-frequency convention.

For initial data $u_0$ on the real line, Fourier transformation or convolution with the fundamental solution gives

For $u_t=D{u}_{xx}$ and $u(x,0)=1_{[-a,a]}(x)$, convolution with the heat kernel gives

For the heat equation $u_t=\frac{1}{2}{u}_{xx}$ on $x>0$ with the Neumann boundary condition $u_x(t,0)=0$, the method of images givesThus $u(t,x)={\int }_0^{\infty }{K}_t^N(x,y)f(y)dy$. This is also the transition kernel of Reflected Brownian motion.

For $u_t=\frac{1}{2}{u}_{xx}$ on $(0,1)$ with initial data $g$ and time-dependent Dirichlet boundary conditions $f_1,f_2$, stop Brownian motion at its first hit of $0$ or $1$. Applying Itô formula to $u(t-s,B_s)$ yields

For a linear evolution equation with zero initial data, Duhamel's principle integrates the homogeneous propagator against the forcing time. For the forced heat equation,

For the planar heat equation with point injection of strength $p(s)$, zero initial data gives $\Psi (t,\mathbf{r})={\int }_0^{t}p(s)e^{-r^2/[4D(t-s)]}/[4\pi D(t-s)]ds$. If injection is $p_0$ until $t_0$ and zero later, then for $t>t_0$ this equals $p_0[\mathrm{Ei}(-b/(t-t_0))-\mathrm{Ei}(-b/t)]/(4\pi D)$, where $b=r^2/(4D)$. The exponential integral expansion gives $p_0\ln [t/(t-t_0)]/(4\pi D)$ when $b/(t-t_0)\ll 1$.

On the real line, initial data $\delta (x-2\sqrt{D})$ and a source $-A\delta (x+2\sqrt{D})\delta (t-1)$ produceAt $x=0,t=2$, the two terms cancel exactly for $A=\sqrt{e/2}$.

The constant-coefficient one-dimensional advection-diffusion equation isTranslation to coordinates moving at speed $a$ reduces it to the heat equation.

For a passive scalar on a half-line, impose constant total solute flux at its inlet, with initially zero concentration and constant positive advection speed $U$ and mass diffusivity $\mathcal{D}$. A Laplace transform in time givesWith $\xi =Ux/\mathcal{D}$, $\vartheta =U^{2}t/\mathcal{D}$ and $z_{\pm }=(\xi \pm \vartheta )/(2\sqrt{\vartheta })$, its inverse isThe complementary error function describes the smeared front. This is a flux boundary condition, not the constant-concentration inlet solution. At large axial Péclet number, both have the leading advancing front $\frac{1}{2}\mathrm{erfc}(z_{-})$.

The half-line solution of $C_t+U{C}_x=D{C}_{xx}$ with initially zero concentration, unit inlet concentration for positive time, and decay at infinity at fixed time. Both complementary error functions are needed to satisfy the inlet exactly. A Laplace transform gives $\tilde{C}=p^{-1}{e}^{(U-\sqrt{U^2+4Dp})x/(2D)}$. A finite layer requires an additional downstream boundary condition.

For the advection-diffusion equation $u_t=u_{xx}+2a{u}_x$, the multiplication $w=e^{ax}u$ gives $w_t=w_{xx}-a^{2}w$. A Dirichlet boundary condition remains a prescribed value, multiplied by $e^{ax}$ at the endpoint. This real exponential conjugation removes the first derivative without changing a bounded spatial interval.

After the Dirichlet gauge transform for constant drift, put $q=\sqrt{p+a^2}$. The Dirichlet Green function of $p-{\partial }_x^2+a^2$ isThe original unweighted resolvent kernel is $e^{-ax}{R}_q(x,y)e^{ay}$. Its poles are $p=-a^2-(n\pi /L{)}^2$. The kernel is even in $q$, so the apparent square-root branch point is removable.

On $z>0$, consider $p_t+a{p}_z=D(z{p}_z{)}_z$ with $a,D>0$, zero scalar flux $ap-Dz{p}_z$ at the endpoints for $t>0$, and unit initial impulse at the origin. Put $r=a/D$ and $\eta =z/(Dt)$. The normalized similarity solution isThe ansatz $p=(Dt{)}^{-1}f(\eta )$ gives $[\eta f^{\prime }+(\eta -r)f{]}^{\prime }=0$. Endpoint decay sets this integrated constant to zero and hence $f\propto {\eta }^r{e}^{-\eta }$; the gamma function normalizes its integral to one. Its scale is $Dt$, so it converges weakly to a unit impulse as $t\downarrow 0$. Its gamma distribution shape is $1+r$, its maximum occurs at $z=at$, and its expected value is $(a+D)t$. For width proportional to $z$, the associated horizontally averaged plume concentration is proportional to $z^{r-1}{e}^{-z/(Dt)}$: its interior maximum is $(a-D)t$ when $a>D$, its supremum occurs at the origin when $a=D$, and it is singular there when $0<a<D$. The physical finite source regularizes this ideal-origin behaviour.

Writing $u=e^{-\beta x/2}v$ transforms $u_t=u_{xx}+\beta u_x$ into $v_t=v_{xx}-{\beta }^{2}v/4$. A prescribed coordinate derivative $u_x=g$ at an endpoint becomes $v_x-(\beta /2)v=e^{\beta x/2}g$ there. Thus a Neumann boundary condition becomes a Robin boundary condition. The signs are coordinate-derivative signs at both endpoints; outward normal derivatives require a sign change at the left endpoint.

On $(0,L)$, the differential operator $\mathcal{L}={\partial }_x^2+\beta {\partial }_x=e^{-\beta x}{\partial }_x(e^{\beta x}{\partial }_x)$ with homogeneous Neumann boundary conditions is self-adjoint in the weighted inner product with weight $e^{\beta x}$. Its eigenfunctions and nonnegative decay rates areTheir squared norms are $N_0=(e^{\beta L}-1)/\beta$ and $N_k=L[1+{\beta }^2/(4q_k^2)]/2$. The Sturm-Liouville eigenfunction expansion givesThis kernel acts against the measure $e^{\beta y}dy$, not Lebesgue measure alone. It is symmetric in $x,y$; the transition density against Lebesgue measure is $e^{\beta y}{K}_{\beta }(x,y,t)$.

For $L>0$, constant $\beta >0$ and $\mathrm{Re}p>0$, on $(0,L)$ put $b=\beta /2$, $\kappa =\sqrt{p+b^2}$ with the principal square root, andThese satisfy $(p-{\partial }_x^2-\beta {\partial }_x)u_p=(p-{\partial }_x^2-\beta {\partial }_x)v_p=0$ and the left and right homogeneous Neumann boundary conditions, respectively. The weighted Wronskian givesThe first derivative jumps by $-e^{-\beta y}$, so the resolvent integrates against $e^{\beta y}dy$. Its time Laplace transform interpretation is ${\mathcal{R}}_p={\int }_0^{\infty }{e}^{-pt}{K}_{\beta }(t)dt$. The factor $p$ records the zero eigenvalue and constant stationary eigenfunction.